Mixing, transport and turbulence modulation in solid suspensions : study and modelling François Laenen

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François Laenen. Mixing, transport and turbulence modulation in solid suspensions : study and modelling. Other [cond-mat.other]. Université Côte d’Azur, 2017. English. ￿NNT : 2017AZUR4010￿. ￿tel-01534441￿

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Université Côte d’Azur École doctorale de Sciences Fondamentales et Appliquées

Unité de recherche : Laboratoire Joseph-Louis Lagrange – UMR 7293

Thèse de doctorat Présentée en vue de l’obtention du grade de Docteur en Sciences de Université Côte d’Azur

Discipline : Physique

Présentée et soutenue par François Laenen

Mixing, transport and turbulence modulation in solid suspensions Study and modelling

Dirigée par Jérémie Bec, Directeur de Recherche CNRS, Observatoire de la Côte d’Azur et codirigée par Giorgio Krstulovic, Chargé de Recherche CNRS, Observatoire de la Côte d’Azur

Soutenue le 24 février 2017

Devant le jury composé de :

Sergio Chibbaro Maitre de Conférence HDR, Université Pierre et Marie Curie Rapporteur Romain Volk Maitre de Conférence HDR, ENS Lyon Rapporteur Guido Boffetta Professeur, Université de Turin, Italie Examinatrice Aurore Naso CR CNRS, Ecole centrale Lyon Examinatrice Mikhael Gorokhovski Professeur, Ecole centrale Lyon Examinateur Emmanuel Villermaux Professeur, Aix-Marseille Université Examinateur Jérémie Bec DR CNRS, Université Côte d’Azur Directeur Giorgio Krstulovic CR CNRS, Université Côte d’Azur Co-directeur ii

Summary

The transport of particles by turbulent flows is ubiquitous in nature and industry. It occurs in planet formation, plankton dynamics and combustion in engines. For the dispersion of atmospheric pollutants, traditional predictive models based on eddy diffusivity cannot accurately reproduce high concentration fluctuations, which are of primal importance for ecological and health issues. The first part of this thesis relates to the dispersion by turbulence of tracers continuously emitted from a point source. Mass fluctuations are characterized as a function of the distance from the source and of the observation scale. The combination of various physical mixing processes limits the use of fractal geometric tools. An alternative approach is proposed, allowing to interpret mass fluctuations in terms of the various regimes of pair separation in turbulent flows. The second part concerns particles with a finite and possibly large inertia, whose dispersion in velocity requires developing efficient modelling techniques. A novel numerical method is proposed to express inertial particles distribution in the position-velocity phase space. Its convergence is validated by comparison to Lagrangian measurements. This method is then used to describe the modulation of two-dimensional turbulence by large-Stokes-number heavy particles. At high inertia, the effect is found to be analogous to an effective large-scale friction. At small Stokes numbers, kinetic energy spectrum and nonlinear transfers are shown to be modified in a non trivial way which relates to the development of instabilities at vortices boundaries. R´esum´e

Le transport de particules par des ´ecoulements turbulents est un ph´enom`ene pr´esent dans de nom- breux ´ecoulements naturels et industriels, tels que la dispersion de polluants dans l’atmosph`ereou du phytoplancton et plastiques dans et `ala surface des oc´eans. Les mod`elespr´edictifs classiques ne peuvent pr´evoir avec pr´ecision la formation de larges fluctuations de concentrations. La premi`erepartie de cette th`ese concerne une ´etude de la dispersion turbulente de traceurs ´emis `apartir d’une source ponctuelle et continue. Les fluctuations spatiales de masse sont d´etermin´ees en fonction de la distance `ala source et `al’´echelle d’observation. La combinaison de plusieurs ph´enom`enes physiques `al’origine du m´elange limite la validit´e d’une caract´erisationde g´eom´etrie fractale. Une approche alternative est propos´ee, permettant d’interpr´eter les fluctutations massiques en terme des diff´erents r´egimes de s´eparation de pair dans des ´ecoulements turbulents. La seconde partie concerne des particules ayant une inertie finie, dont la dispersion dans l’espace des vitesses requiert de d´evelopper des techniques de mod´elisationadapt´ees. Une m´eth- ode num´erique originale est propos´ee pour exprimer la distribution des particles dans l’espace position-vitesse. Cette m´ethode est ensuite utilis´eepour d´ecrirela modulation de la turbulence bi-dimensionnelle par des particules inertielles. A grand nombres de Stokes, l’effet montr´eest ana- logue `acelui d’une friction effective `agrande ´echelle. Aux petits Stokes, le spectre de l’´energie cin´etique du fluide et les transferts non-lin´eaires sont modif´ees d’une mani`ere non triviale. Contents

1 Introduction and context 1

2 Definitions and concepts 11 2.1 Navier–Stokes equations ...... 11 2.1.1 Structure functions and intermittency ...... 12 2.2 Turbulence in two dimensions ...... 15 2.2.1 Two dimensional Navier–Stokes equations ...... 15 2.2.2 The double cascade framework ...... 17 2.2.3 Energy and enstrophy budgets ...... 21 2.3 Relative dispersion rates of Lagrangian trajectories ...... 22 2.3.1 Lyapunov exponents ...... 22 2.3.2 Separation rates ...... 23

I Turbulent dispersion and mixing 27

3 Tracers dispersion in two dimensional turbulence 29 3.1 Introduction ...... 29 3.1.1 Diffusion at long times ...... 33 3.1.2 Continuous source ...... 36 3.2 middling version ...... 36 3.2.1 Fluid phase integration ...... 36 3.2.2 Injection mechanism ...... 38 3.2.3 Removal mechanism ...... 39 3.3 Results ...... 39

iii iv CONTENTS

3.3.1 One point dispersion ...... 39 3.3.2 Two-point correlation ...... 43 3.3.3 Phenomenological description ...... 49 3.4 Brief conclusion ...... 54

II Inertial particle-laden flows 55

4 A lattice method for the numerical modelling of inertial particles 57 4.1 Inertial particles dynamics ...... 57 4.1.1 Individual particles ...... 57 4.2 The modelling of dispersed multiphase flows ...... 61 4.2.1 From microscopic description to macroscopic quantities ...... 62 4.3 Description of the method ...... 67 4.4 Application to a one-dimensional random flow ...... 70 4.4.1 Particle dynamics for d =1 ...... 70 4.4.2 Lattice-particle simulations ...... 74 4.5 Application to incompressible two-dimensional flows ...... 79 4.5.1 Cellular flow ...... 79 4.5.2 Heavy particles in 2D turbulence ...... 81 4.6 Conclusions ...... 86

5 Turbulence modulation by small heavy particles 89

6 Conclusions and perspectives 113 6.1 Turbulent transport of particles emitted from a point source ...... 113 6.2 Modelisation of small inertial particles ...... 115 6.3 Turbulence modulation by small heavy particles ...... 116

Appendices 119

A Software details 121 A.1 GPU2DSOLVER ...... 121 A.1.1 Numerical scheme ...... 121 A.1.2 Forcing ...... 123 A.1.3 Parallelisation ...... 124 A.2 LAGSRC2D ...... 126 A.2.1 Numerical implementation ...... 126 A.2.2 Injection rate ...... 127 A.3 LOCA: Lattice model for heavy particles ...... 128 A.3.1 Finite volume fluxes ...... 128 A.3.2 Dynamic grid resizing (DGR) ...... 129 CONTENTS v

A.3.3 Parallelisation ...... 131

B SoAx: a convenient and efficient C++ library to handle simulation of heterogeneous particles in parallel architectures 133 Acknowledgments ...... 159 vi CONTENTS Nomenclature

V Tracer velocity.

Vp Inertial particle velocity.

X Tracer position.

Xp Inertial particle position. xS Position of the injecting source.

δru Longitudinal velocity increment between two particles separated by a distance r.

✏ Total energy dissipation rate.

✏↵ Energy dissipation rate due to large-scale friction.

✏⌫ Energy dissipation rate due to viscous friction.

 Molecular diffusion coefficient.

T Turbulent diffusion coefficient.

V Diffusivity in the velocity space.

φm Massic ratio between the solid and the fluid phase.

−1 φS Source injection rate. Units are T .

φv Volumic fraction of the particles.

Fluid stream function.

vii viii CONTENTS

⇢(x) Fluid density field.

⇢p(x) Particle density field. ⇢p(x)=np(x)Mp where Mp is the mass of one particle. Units −d of ⇢p are ML , where d is the space dimension. urms Root-mean square value of the velocity field, averaged over all components. a Particle radius. d Dimension of the position space. mp Total mass of the particle population. mQL Quasi-Lagrangian mass.

−d np(x) Particle number density field. Units are L , where d is the space dimension. p df Probability density function.

Rmax Radius of the circle around the injecting source beyond which particles are removed from the domain.

St Stokes number.

TCL Lagrangian velocity autocorrelation time. Z Fluid enstrophy. CHAPTER 1

Introduction and context

This thesis is to be put in the context of the ERC research project Atmoflex, running from 2010 to 2014, from which it was partly funded. This project aimed at providing a better understanding of fluctuations in particulate transport and mixing processes in turbulent flows, such as pollutants in the atmosphere or scalar fields like salinity in the oceans, or merging density-matched gases.

Turbulence: a multi-scale phenomenon

Turbulence is a phenomenon characterised by a chaotic and out of equilibrium state of a physical non-linear system. It may be found in numerous situations and applications, such as non-linear optics, passive advection in fluids, waves interactions at water surface, magnetic dynamos, etc. All these processes involve a system in which energy is exchanged between many degrees of freedoms. Some theories emerged during the last century trying to find rigorous mathematical formulation of the phenomenological predictions with the use of non-equilibrium statistical mechanics, but an unifying and universal theory is still lacking. In this thesis is considered the Navier–Stokes turbulence, recognisable by the broad range of vortex sizes it generates in fluids. It has the particularity of re-distributing energy injected at a scale L down to smaller and smaller scales until molecular dissipation stops this cascading energy transfer. This transfer can be thought of as a pipe carrying energy in the scale space. It takes place in a range of scales, called inertial range, in which statistics of the flow are believed to be independent of the way it is forced and dissipated. This transfer arises because of the non-linear term in the Navier–Stokes equations which makes this equation unsolvable exactly.

1 2 CHAPTER 1. INTRODUCTION AND CONTEXT

A striking characteristic of turbulent flows is the apparent chaotic trajectories of the species they transport. This manifests into the unpredictability of the position of solid grains in suspensions, or concentration values downstream of an emitting source of pol- lutant. The same observation holds for the simultaneous transport of multiple objects. Consider for example placing two buoys on the surface of a river in a turbulent state. ini- tially very close to each other, then compare their trajectories. Whatever small are their initial separation, they are likely to diverge in a finite time, demonstrating the chaotic nature of turbulence through the sensibility to initial conditions. Even though chaos theory underwent tremendous interest since the beginning of the 20th century, the statistics of the velocity differences between two points in space in the inertial range lead to pair separation rates different from what can be predicted by chaotic motions. This mixing of separation regimes between particles lead to even more complicated prediction about their average concentration with possibly high fluctuations. In addition, there still subsists a lack of understanding regarding the universality of the fluid velocity statistics, i.e. the independence with respect to the forcing and dissipation mechanisms. One is then forced to treat virtually each situation as a case-by-case study.

Lagrangian and Eulerian description

Considering the release of a cloud of a given substance, such as dye or passive pollutant, in the atmosphere or in the ocean, one may ask multiple questions: how will its shape be deformed with time? What would be the maximum expected concentrations? What about the strength of its variations in space and time ? What are the probabilities that a concentration grows above a given threshold and how often? These questions are naturally raised by health and environmental issues (Shi et al., 2001). Indeed, it is often required to predict concentration levels of various constituents, like ash, particulate matter, radioactive elements, etc. Respective examples are volcanic eruptions and their impact on air traffic(Sch¨afer et al., 2011), road traffic regulation in pe- riods of intense air pollution (Han & Naeher, 2006), intra and intercontinental radioactive transport (Wotawa et al., 2006), etc. Living organisms are also concerned by large fluctua- tions: for the ones following concentration gradients (chemotaxis), such as moths attracted by male pheromones, (Mafra-Neto et al., 1994), large scale-induced density fluctuations make this approach much more difficult. Furthermore, the multi-scale property of turbulent flows may be appreciated into its self-similar character: irregular patterns made by smoke coming out of a cigarette resemble the ones escaping from a small house chimney, or a large industrial power plant, or even, to some extents, from a volcano. Another noticable effect of turbulent transport is the fact that particles may get trapped inside vortices, leading to higher concentration values. Ac- tually, it is known that the probability density function of the passive scalar concentration have tails decreasing slower than a Gaussian distribution (Warhaft, 2000). Such trapping events may then be related to regions of the flow with given topological properties (see, for 3 instance, Bhatnagar et al. (2016)). Figure 1.1 shows some examples of systems concerned with transport of solid particles or continuous fields . When measuring the spatial and temporal properties of the transported species, one can choose to adopt two main points of view. The first one is Lagrangian (Lin et al., 2011). It forces the description of phenomena in a framework attached to the transported particle. Lagrangian models are very useful for various reasons. They offer a more natural way to model turbulent transport of solid particles. In addition, they suffer from only infinitesimal numerical diffusion, allowing to recover strong concentration gradients. They are also more numerically stable and allow for bigger time steps. Furthermore, the inverse Lagrangian transport may be used to track sources of contaminants or green house gases (Trusilova et al., 2010). Finally, computational resources available nowadays allow to routinely sim- ulate systems with millions of particles, which is sufficient in some application, and was undoable a decade ago. Another way to measure turbulent transport is by considering a continuous concentra- tion field of a given released substance. We then talk about Eulerian description. This point of view considers a fixed spatial grid on which are defined tensorial quantities, mostly scalars, like concentration. Although Eulerian framework suffers from numerical diffusion and instabilities, it offers a more natural description of a field transported by the under- lying carrier flow, and allows for more convenient way to model back reaction from the substance on the fluid. Eulerian formulations also offer more handy ways to parallelise nu- merical codes, especially when considering domain decomposition among a large number of computational nodes.

The challenge of numerical simulations

In order to perform fundamental studies regarding all questions risen above, computers have played a growing role into fluid mechanics since their invention in the middle of the 20th century, although methods of discrete calculation to resolve the fluid equations already appear in the pioneering works of Richardson (see Richardson (2007) (reprint) and reference in Hunt (1998)). First computations based partly on these works appeared during the 40s using ENIAC and the first three-dimensional simulations were later performed Los Alamos laboratory (Harlow, 2004). Since then, more and more power was dedicated to simulate flows as realistic as possible. One could ask why tremendous supercomputer that we see today in 2016 are still not sufficient ? This is again due to the multi-scale nature of turbulence but also to its temporal fluctuation. Indeed, turbulence is characterised by different eddy sizes and time scales. The width of the spatial scale separation can be measured by the adimensional Reynolds number Re, which is the ratio between advection strength UL and viscosity . The larger Re, the wider the scale separation. Furthermore, the complexity of numerical calculations, or number of degrees of freedom, grows as Re9/4, which translates into a rapid increase of the need for computational resources. 4 CHAPTER 1. INTRODUCTION AND CONTEXT

(a) Amazonia forest fed with transported (b) Surface sea temperature along with sand from Sahara desert. See Yu et al. concentration spots of radioactive Cesium (2015). 134 following Fukushima eruption. Credit: WHOI

(c) 3D simulation of ash spreading following (d) Phytoplankton bloom off the Iceland the Calbuco eruption in April 2015 combined coast. Credit: NASA. with actual data from Suomi NPP satellite. Credit: NASA.

Figure 1.1: Examples of relevant issues involving turbulent transport.

Turbulent motions often arise above a critical Reynolds number, of the order of a few thousands, depending on the system. Very high Reynolds numbers are especially found in the case of planetary-scale motions, where the separation in scales spans from millimetres to kilometres, Re can reach 108. For a moving car at 90 km/h, it reaches 106. Furthermore, because of its inherent intermittent character (see section 2.1.1), i.e. with velocity statistics displaying tails much broader than Gaussian, turbulence is also characterised by extreme events. These can represent up to 105 times the mean value. They are not so probable (hence the name extreme) but sufficiently to play a key role in the dynamics, a property shared with other strongly non-linear phenomena. Because of these extreme events, numerical simulations have to be run for a very long time and / or at very large resolutions. State of the art three-dimensional turbulence numerical simulations achieve Reynolds 5 numbers up to 45000, with a Taylor microscale Reynolds of 1300 for a resolution of 81923 grid points (Yeung et al., 2015). How to overcome this numerical challenge and to reduce its complexity ? The answer is not to resolve explicitly all the scales. Tons of scientist and engineers have worked in this direction, leading to very clever and tricky methods and mathematical models to represent the turbulent motions at scales smaller than the one of interest. Such activities are referred to as sub-grid scales modelling (Majda & Kramer, 1999; Pope, 2000). Such small scale motions however play a very important role, impacting larger ones in situations like clouds (Bodenschatz et al., 2010), or planets (Armitage, 2015) formation. Careful design of the models is thus required, and these must be continuously improved based on a better knowledge of microscopic phenomena. Particle transport is a typical situation where such problems occur. Traditional esti- mations used in the mechanical and environmental engineering communities are based on mean-field approach (Opper & Saad, 2001): substances are advected by larger scales (big eddies) and the effect of smaller scales are just perturbations, treated for example as addi- tional sources of diffusion, called eddy-diffusivity. Even if we know that the effect of small scales motions on the larger ones cannot be accurately represented by a simple diffusion operator (Corrsin, 1975). These models yield correct predictions for concentrations that are far from an emitting source or during long time averages. This topic is itself enclosed into a much wider research area, multiphase flow modelling, dealing with the numerical simulation of multiple species simultaneously present in a spatial domain, interacting or not. An example of application is the prediction of the fluid regimes transition (Labourasse et al., 2007; Monahan & Fox, 2007; Van der Hoef et al., 2008), when one wishes to test how a device such as a fluidized-bed reactor will scale when going from the laboratory to the power plant (Ge et al., 2007).

When back-reactions come at play

Under some circumstances, the transported phase may have a significant impact on the carrier flow, which may be desirable or not. Such effects are called two-way coupling, and induce non-linear effects that can hardly be predicted by phenomenological arguments. In the case of dispersed solutions, several application are worth mentioning. Consider the example of fluid transportation in a pipe. Pressure losses in such conducts lead to very high power consumption. It is thus of economical interest to try to reduce energy dissipations in such flows. This dissipation attenuation was actually observed in suspensions of certain additive types. Fibrous additives, like polymers (e.g. nylon, cotton) have shown to reduce drag efficiently (White & Mungal, 2008; Yang, 2009) though the situation is not yet clear for non fibrous materials, like rigid bodies of various shapes (spherical or platelet, needle-shaped...) and size. The interplay between suspended solids and the liquid phase is also a process of prime importance for planet formation (Barranco & Marcus, 2005). Indeed, gas giant planets have been shown to migrate, i.e., to form far 6 CHAPTER 1. INTRODUCTION AND CONTEXT from their star and come closer at later stages, with the corresponding mechanisms depend on their size. The modelling of these back reactions leads to additional challenges. Indeed, when using Lagrangian description, one needs to reconstruct a force field to act on the carrier phase, and losses of convergence and precision occur a this step. When available, an Eulerian point of view for the suspended phase is thus generally preferable.

The case of two dimensional turbulence

Turbulent flows that are considered in this thesis are two dimensional. Actually, there is no such a thing as a real two-dimensional flow in nature, so why bother studying them ? In some situations, a flow which is a priori three dimensional may exhibit 2D turbulence dynamics. A particular example is the case when swirling motions are physically constrained to evolve in a thin layer. These constrains may take various forms. They can be created through the formation of a thin layer by geometric confinement with solid boundaries, or by the presence of strong rotation (Pouquet et al., 2013) and / or stratification, in which case the formation of two dimensional layers called pancakes is observed (Godoy-Diana et al., 2004). Some systems may combine all these effects. Examples are large-scale geophysical flows in the Earth atmosphere and oceans (Nastrom & Gage, 1985; Monin & Ozmidov, 1985). One way to reproduce such constrains in laboratory experiments is to isolate the fluid within a thin tank. One can then use electrically conducting fluid put into motion by an array of magnets (Paret et al., 1999; Boffetta et al., 2005). Soap films are also a really good ”planetary toy” because this setup allows to reproduce turbulence under gravity in a thin spherical layer combined with rotation (Kellay et al., 1998; Rutgers, 1998; Seychelles et al., 2010). Another notable example of flows dimensionally constrained are the protoplanetary disks (Barranco & Marcus, 2005) where tall columnar vortices form. In gaseous nebulae, both processes of planetary formation and migration depend on the vortices structure. Accretion probabilities and transport will indeed vary depending if the flow is dominated by turbulent eddies and long-lived coherent vortices. One should recover three dimensional turbulence phenomenology when considering the constrained flow at sufficiently small scales, or when removing the constrains. For example, dimensionality may be measured as a function of the rotation or stratification intensity (Smith et al., 1996; Deusebio et al., 2014; Sozza et al., 2015). This mechanisms at transition between 2D and 3D is called bidimensionalisation and is also a very interesting research topic by itself.

All the notional concepts described above form the core part of this thesis work. How does the temporal and spatial correlations of fluid velocity in two dimensional turbulence impact mass distribution when it is initially released from a limited region in space ? How 7 is the turbulence affected by small inertial particles ? How to represent such particles in term of a field in a way that is computationally affordable and physically correct ? This manuscript is organised in two parts. The first one is aimed at describing how mass continuously injected from a point source in two-dimensional turbulence fills the space, targeting situations such as oil spreads at ocean surface. Lagrangian particles are emitted via numerical simulations, and mass fluctuations are quantified using fractal dimension and a novel description based on relative pair separation. The second part is split in two chapters. The first one describes a numerical approach for the simulation of small heavy particle suspensions in two-dimensional turbulence. It treats the kinetic equation associated to the dynamic on a regular lattice in position space and finite volume method in velocity space. The second chapter shows an application to the study of turbulence modulation by small and heavy particles. The modifications of large and small scales statistical quantities of the fluid are assessed. In order to realise these studies, scientific libraries have been developed for massively parallel computations using GPGPUs. These libraries were used to simulate the two dimen- sional turbulence from Navier–Stokes equations as well as particle emission and dynamics. Some details about their implementation are explicited in the appendix A. Another library to handle systems of large number of particles was jointly developed. It was aimed to ease the process of creating particles with various properties like mass, electric charge, etc. while keeping performance when simulating their dynamic on various parallel architectures (see B). The two parts may be read independently, with some concepts being introduced in chapter 2.

First part: Tracers dispersion from a point source

One can have the intuitive picture that on average, an emitted puff of a suspended substance will regularly grow under the effect of diffusion and spread uniformly in space. However, the temporal correlation of the turbulent eddies at all scales bringing together regions of very different concentration values results in creating strong inhomogeneities and gradients. In chapter 3, a system of tracer particles continuously emitted from a point source is studied. The additional challenge compared to traditional turbulent mixing lies in the joint effect of spatial as well as temporal correlations in the particles trajectories. The interplay of these correlations is one of the major issues in turbulence. Only in very few models for the carrier fluid these correlations can be analytically treated, like in the Kraichnan ensemble where temporal correlations are fully disregarded (Celani et al., 2007). High resolution direct numerical simulations of inverse turbulent energy cascade are carried, and the issue of measuring the spatial fluctuations of the particle distribution is addressed. To this end, we propose a phenomenological description which allows us to relate the concentration fluctuations along particle trajectories (quasi-Lagrangian mass scaling) with the tracers 8 CHAPTER 1. INTRODUCTION AND CONTEXT

Figure 1.2: Gaussian concentration distribution (colour map) predicted by mean-field ap- proaches do not allow to reproduce fine structures and high concentration levels of trans- ported particles (black dots). The red cross represents the emitting source. relative dispersion regimes. The idea is to follow an emitted line of particles to quantify its foldings and see how it contributes to the quasi-Lagrangian mass scaling as a function of the distance from the source.

Second part: Modelling particle-laden flows and two-way coupling

Challenges of inertial particles statistical modelling

This second part starts with chapter 4 which addresses the issue of modelling heavy- particle-laden turbulent flows. The dynamics of such particles is first introduced, and the challenge is stressed to provide a correct modelling of this kind of suspensions, which basically lies in the capacity of the particles to form caustics. The adopted mathematical model must then be able to resolve the velocity dispersion of the particle population. A short review of multiphase models dealing with solid suspensions is also presented. A novel numerical method is then introduced. Its originality lies in the absence of any form of closing of the kinetic equation associated with the dynamic. This Liouville equation is integrated explicitly in the phase space. Numerical and physical convergence are assessed, and it is shown that the method reproduces with good accuracy the particle distributions obtained via Lagrangian direct simulations. 9

Turbulent modulation by small heavy particles Chapter 5 presents a study about the impact of back-reaction of small heavy particles on a two-dimensional turbulent flow. A short overview of these effects is first presented and the chosen models representing the considered Stokes number asymptotics are introduced. Di- rect numerical simulations of direct enstrophy cascade are performed, varying the particles mass load. The effect of particles on various statistical properties of the flow is assessed in the asymptotics of low and large Stokes numbers. The modification of global quantities is measured, such as mean energy and enstrophy, as well as the modifications of the scaling in the velocity field through non-linear transfers and dissipations. Impacts on small-scale statistics is also addressed, in particular the modification of intermittency modification. Finally, we also measured how the particles preferential concentration property is affected. 10 CHAPTER 1. INTRODUCTION AND CONTEXT CHAPTER 2

Definitions and concepts

Contents 2.1 Navier–Stokes equations ...... 11 2.1.1 Structure functions and intermittency ...... 12 2.2 Turbulence in two dimensions ...... 15 2.2.1 Two dimensional Navier–Stokes equations ...... 15 2.2.2 The double cascade framework ...... 17 2.2.3 Energy and enstrophy budgets ...... 21 2.3 Relative dispersion rates of Lagrangian trajectories ...... 22 2.3.1 Lyapunov exponents ...... 22 2.3.2 Separation rates ...... 23

Concepts and mathematical tools relevant to each chapter will be introduced in their respective opening. As all chapters of this thesis share in common the framework of two- dimensional, incompressible turbulence, the characteristics of such flows are highlighted and compared to their three-dimensional equivalent in this chapter. Some concepts and notations regularly appearing throughout this manuscript are also introduced.

2.1 Navier–Stokes equations

An incompressible velocity field u(x,t) at position x is described by the following equations: @ u +(u )u = p + ⌫ 2u + F , (2.1) t · r −r r u =0. (2.2) r ·

11 12 CHAPTER 2. DEFINITIONS AND CONCEPTS

⌫ is the kinematic viscosity, the term ⌫ 2 being responsible for the dissipation of large r gradients at small scales. p is the pressure which ensures the incompressibility condition (2.2). F denotes the external forcing that maintain the velocity field in a statistically steady, developed turbulent state. In the absence of dissipation and forcing, i.e. when ⌫ = 0 and F = 0, several quantities, or invariants characterise the flows. In three dimensions, global invariants are energy E = u 2 and helicity H = ω u . The averaging operation is taken over space and k k h · i time inD theE statistically stationary regime. Properties of the flow may be assessed through the rate-of-strain tensor, a second order tensor encompassing the gradients of each velocity component u1,2,3, namely:

@1u1 @2u1 @3v1 = ru = @ u @ u @ v . A 0 1 2 2 2 3 21 @1u3 @2u3 @3v3 @ A This matrix, like any, can be decomposed into a symmetric and an antisymmetric part, which are often named Ω and S, respectively:

1 1 Ω = ru (ru))T , S = ru +(ru))T . (2.3) 2 − 2 h i h i Ω = 1 (@ u @ u ) are the vorticity components, and S = 1 (@ u + @ u ) the shear ij 2 i j − j i 2 i j j i components. From these tensors, a criterion can be built to determine whether locally in space the flow is dominated by shear or vorticity. In two dimensions, this criterion is given by the Okubo-Weiss criterion = Ω2 S2 (Okubo, 1970; Weiss, 1991). Due to the Poisson W − equation for pressure 2p = Ω2/2 S2, the following relation holds for incompressible r − flows: Ω2 =2 S2 . In three dimensions, one can use the Q R criterion. The matrix has three invariants ⌦ ↵ ⌦ ↵ − A under canonical transformations: P = Tr(A), Q = Tr(A2/2), R = Tr(A3/3). The − − determinant of the characteristic equation for is then given by ∆ = (27/4)R2 + Q3 and A the Q R plan defines 4 regions corresponding to different combinations of eigenvalues − of . Vorticity dominates for large positive ∆ with vortices that are either compressed A (R<0) or stretched (R>0). On the contrary, strain will dominate for ∆ < 0. See Cantwell (1993) for more details.

2.1.1 Structure functions and intermittency The statistics of velocity differences between two points separated by a distance r constitute an important quantity in turbulent flows. Their probability density function gives an important idea about collision probabilities or particles separation rate (see section 2.3). The scaling of their moments also gives information about the scale-invariance of turbulent 2.1. NAVIER–STOKES EQUATIONS 13

flows. Velocity differences, or increments, are defined by:

δ u = u(x) u(x + r) . (2.4) r h − i They constitute a random quantity. When considering homogeneous, isotropic flows, as is the case in this manuscript, δru only depends on the modulus of r and is independent of the position x. The structure functions are defined by its moments of various order:

k q Sq(r)= (δr u) (2.5)

k r D E where δr u = δ u(r) is the longitudinal component of δ u. Alternatively, one may r · r r also consider S? using the projection of δu(r) on directions orthogonal to r. The average in (2.5) is taken over the whole space and over time series in the statistically steady regime. In a developed turbulent regime, Sq(r) follows a power-law function of the scale r: S (r) r⇣(q) (2.6) q / The scaling behaviour is the following. Suppose a large scale forcing at lI of the flow which is in a statistically stationary state. At scales smaller than lI , statistical quantities can be assumed to be homogeneous. This is the inertial range. This regimes goes down to a scale ld, the dissipative scale, at which energy is dissipated. In 1941, Andrei Kol- mogorov made a series of hypothesis that leads to quantitative predictions about velocities increments δru and ld. One is that the energy dissipation rate ✏ > 0 (we choose here to con- sider this quantity as positive) has a non-zero limit at vanishing viscosity (⌫ 0). This is ! called dissipative anomaly. Another of his hypothesis, shared with Onsager and Heisenberg, sometimes called universality assumption, is that all the small-scale statistical properties are uniquely and universally determined by the scale r and the energy dissipation rate ✏. −1 Given the dimensions of the relevant quantities at play in the inertial range, [δru]=LT , [✏]=L2T −3 and [r]=L, an expression for ✏ follows from dimensional analysis:

(δ u)3 ✏ r . (2.7) ⇠ r The Reynolds number associated to the scale r reads:

(δ u)r ✏1/3r4/3 Re(r) r , (2.8) ⇠ ⌫ ⇠ ⌫ which yields an expression for the dissipative scale corresponding to Re(ld)=1:

3/4 −1/4 ld = ⌫ ✏ (2.9)

Note that it is possible to derive a formal expression for S3. Multiplying the Navier- Stokes equations (2.1) for u(x) by u(x + r) yields an equation for the time evolution of 14 CHAPTER 2. DEFINITIONS AND CONCEPTS

S2 as a function of S3, called Karman-Howarth, or simply KH relation. It may then be solved for S3 in the statistically stationary state (for details of the calculations, see Frisch (1995); Landau & Lifshitz (1987)), yielding the only exact result in turbulence known as the 4/5-law (Kolmogorov, 1941): 4 S (r)= ✏r. (2.10) 3 −5 This relation implies that there is a constant energy flux in the inertial interval of scales, equal to the one injected at the stirring scale. It also displays that the velocity increments are negatively skewed: particles getting closer are more probable than particles separating. Kolmogorov then assumed strict self-similarity for the velocity differences, i.e. the k h k existence of a unique exponent h such that δ u λ δr u. This implies that ⇣(q)=hq is λr ⇠ a linear function of q. Requiring (2.10), one gets h =1/3 hence:

q/3 Sq(r)=Cq(✏r) , (2.11)

where the Cp’s are universal dimensionless constants. Furthermore, The quantity S2 is actually linked to the power-spectra of the velocity. Let us first define the energy density e(k) in the Fourier space (we note the Fourier transform), with F k the wave-vector: 1 uˆ(k) 2 k r e( )= d k k = Rii( ) . (2.12) (2⇡) 2V F " # Xi d V = L is the domain volume and Rij(r)= ui(x)uj(x + r) is the velocity correlation h i ik·r tensor. The following convention for the Fourier transform is used: uˆ(k)= Rd e . The kinetic energy spectrum may then be defined in various forms, such as the two following: R 2 0 0 0 d−1 uˆ(k) E(k)= dk δ( k k)e(k )= dΩkk k k , (2.13) Rd − 2 Z ZΘk � � � � with Θk the hypersphere in Fourier space of radius k and Ωk the solid angle element. Owing to the Parseval theorem, the mean energy in our system can be evaluated either in Fourier or physical space via: 1 1 u(x) 2 E = k k dx = E(k)dk. (2.14) V Rd 2 Z Z0 The following relation gives the equivalence between structure functions and spectra. Given a power-law spectrum:

F (k) k−n, 1

(f(x) f(x + r))2 r n−1. (2.16) − / | | ⌦ ↵ 2.2. TURBULENCE IN TWO DIMENSIONS 15

The energy spectrum is sometimes more easily interpretable than the structure function which is its physical pendant. It is also easily obtainable in numerical simulations that use spectral methods to integrate the velocity fields (see appendix A.1). Using (2.11) and (2.16) for q = 2, one gets

E(k)=✏2/3k−5/3, (2.17) which is also the only dimensionally correct combination of ✏ and k. This spectrum shape has been effectively observed in numerical and experimental works. However, in three dimensions, self-similar hypothesis shows to be more and more inexact as the moment q d log Sq increases (Frisch, 1995). Indeed, the exponent ⇣(q)= d log r has been shown to be a strictly concave function of q, rather than linear. This is called anomalous scaling.

2.2 Turbulence in two dimensions

This thesis work mainly involves two-dimensional turbulent flows, which display some in- teresting features that are phenomenologically different from their three-dimensional pen- dants.

2.2.1 Two dimensional Navier–Stokes equations

In incompressible flows, @iui = 0 so that the two-dimensional velocity field is fully deter- mined from the function (x, y,t) via the relation u = ? =(@ u , @ u ) (the sign of r y x − x y ? may vary in the literature). The level sets of represent the stream-lines, with u being r everywhere tangent to these level curves, hence the name stream-function for . Vorticity is related to by the relation ! = u = 2 . The evolution equation for reads: r⇥ −r 1 @ + , 2 = ⌫ 2 + f 0 (2.18) t 2 { r } r r where denotes the Poisson bracket, or Jacobian, such that f,g = @ f@ g @ f@ g. {} { } x y − x y Kraichnan (1967) already conjectured that energy would accumulate in the gravest mode kmin allowed by the boundary conditions (see below section 2.2.2). This accumulation of energy at large scale would eventually a condensate (illustrated in Figure 2.1), analogous to a Bose-Einstein condensate. In the real world, mechanisms arise from various physical origins to prevent this large scale energy piling up, like Rayleigh friction in stratified fluids or the friction induced by the surrounding air in soap-film experiment. An additional linear term ↵ is often − added in Navier-Stokes equations to represent this friction, so that f 0 = f ↵ . ↵ is − thus the Ekman friction coefficient responsible for the large scale energy dissipation. It is often used in numerical simulations to reach a statistically stationary state, especially in the two-dimensional inverse cascade (Salmon, 1998). The origin of its linear form may 16 CHAPTER 2. DEFINITIONS AND CONCEPTS

Figure 2.1: Vorticity condensate in the two dimensional inverse cascade. From Boffetta & Ecke (2012).

be exemplified the following way: near solid boundaries with no-slip condition, a laminar Poiseuille profile is usually admitted. Say the surface is horizontal at z = h, with z the vertical direction, then the velocity reads u(z)= ↵ (z h)2 and ⌫(@2 + @2 + @2) 2u 2⌫ − x y z r ⇠ ⌫(@2 +@2)u ↵u. In experiments, it is also this drag form that is adopted for liquid friction x y − on a soap film, or with the bottom of a container. Also ion-neutral collisions in ionospheric plasma give rise to such a friction.

The equivalent equation for the vorticity is obtained by taking minus the Laplacian of equation (2.18):

@ ! + u r! = ⌫ 2! ↵! + f (2.19) t · r − !

! is written in non-bold font to explicit that it is treated as a scalar quantity in the 2D plane: the vortex lines are always perpendicular to the flow plane. Equation 2.19, except for the forcing f!, expresses vorticity transport by the flow and dissipation by small-scale molecular viscosity and large-scale friction. One important difference between (2.19) compared to its 3D equivalent is the lack of the vorticity stretching term (ω r)u which is identically 0 · in 2D. In 3D, it is responsible for vorticity amplification in the vortex stretching direction due to angular momentum conservation. 2.2. TURBULENCE IN TWO DIMENSIONS 17

2.2.2 The double cascade framework Global invariants Neglecting the large-scale friction and force terms in (2.19) and multiplying (2.19) by ! and averaging, one gets D Z = ⌫ 2! where t r 1 Z = !2 (2.20) 2 is the enstrophy. Thus, for inviscid flows (⌫ =⌦ 0), enstrophy↵ is conserved along Lagrangian trajectories1. Actually, !nd2x is a constant of motion for all n>0. In two-dimensional turbulence, enstrophy and energy are the quadratic invariants. R To give an intuition about the mechanism at play, the unforced Navier–Stokes equations for the velocity Fourier coefficients are first introduced: k k @ uˆ (k)+ δ i j ip uˆ (p)ˆu (q)= ⌫ k 2 uˆ (k). (2.21) t i ij − k2 j i j − k k i k p q ✓ ◆ =X+ The second term on the left-hand side is the Fourier transform of the non-linear term. It shows that the mode k interacts with modes p and q such that p = q + k, i.e. p, q and k must form a triangle. This is called a triadic interaction, and conserves both energy and enstrophy. In order to grasp the essence of energy transfers in two-dimensional turbulence, one can refer to the paper by Kraichnan (1967) who deduced the direction of the cascades of energy and enstrophy with statistical mechanics arguments. The term cascade refers to the fact that the injected energy (or enstrophy) is transferred at a constant rate through the scales. This transfer results from the non-linear term in (2.1) and takes place until molecular dissipation counter-balances at the dissipation scale. Another phenomenological argument was already advanced by Fjørtoft (1953) to predict the direction (in the scale space) of these transfers. A triadic interaction between wave- numbers k1

The direct enstrophy cascade is considered to be the process responsible for vorticity filaments stretching and folding, creating stronger and stronger vorticity gradients until they are eventually dissipated by molecular viscosity. See (Kraichnan & Montgomery, 1980; Monin & Ozmidov, 1985) for details.

Spectrum scaling in the dual cascade The energy and enstrophy spectra are related through Z(k)=k2E(k). In the inverse cas- cade, under the assumption of Kolmogorov phenomenology, the energy spectrum exponent is identical to the one in three dimensions:

2/3 −5/3 E(k)=C1✏ k , (2.23) 2/3 −1/3 Z(k)=C1✏ k , (2.24) where ✏ is the energy dissipation rate and C1 is a constant, determined to be in the range 5.8 7.0(Paret & Tabeling, 1997). ⇠ − In the direct enstrophy cascade, the spectra are:

2/3 −3 −1/3 E(k)=C2✏! k [ln(k/kf )] , (2.25) 2/3 −1 −1/3 Z(k)=C2✏! k [ln(k/kf )] . (2.26) where ✏! represents this time the enstrophy dissipation rate, analogous to ✏ for energy. kf =2⇡/lf denotes the forcing wave-number corresponding to the forcing length lf . The presence of the logarithmic factor in (2.25) and (2.26) is a correction that ensures that the enstrophy flux is constant across the inertial range (see Kraichnan (1971); Rose & Sulem (1978) for details), This factor is important for regularity reasons. Indeed, the total enstrophy Z = k2E(k)dk u 2 with E(k) k−3 logarithmically diverges, and the ⇠ kr · k / velocity field is not differentiable. It also implies that enstrophy fluxes are less local (i.e., R contributions to the flux at k can come from a much wider range of wave-numbers around k) than their energy pendants. Those scalings for energy and enstrophy spectra have been indeed observed in numerical studies, already in Borue (1994), and experimentally in large-scale geophysical or quasi-2D stratified flows and soap films (Boer et al., 1984; Rivera & Wu, 2000; Daniel & Rutgers, 2002). Figure 2.2, coming from a direct numerical simulation at very high resolution (Boffetta & Musacchio, 2010), illustrates these two regimes. In the inverse cascade, one can also derive an expression for S3(r) in the same way than in three dimensions (see section 2.1). It reads: 4 S (r)= ✏r. (2.27) 3 3

Compared to (2.10), S3 is positively skewed in two dimensions. In the direct cascade, the 1 3 prediction is S3(r)= 8 ✏!r . 2.2. TURBULENCE IN TWO DIMENSIONS 19

100

10-2

10-4 2/3 α ε -6 10 1

E(k)/ A δ 10-8 B 0.1 C -10 10-6 10-5 E 10 ν D 1 101 102 103 104 k

Figure 2.2: Energy spectra of two dimensional flows, obtained via Direct Numerical Sim- ulations for spatial resolution up to 327682 (from Boffetta & Musacchio (2010)). In those simulations, the forcing wave-number is set at kf = 100.

Effect of Ekman friction

Due to the energy flow toward the large scales in the two dimensional inverse cascade, it is necessary to provide a low-k energy sink if ones wants to achieve statistical stationary state. This is why the Ekman friction term is so important in two-dimensional simulations. This term however has some impact on the flow structure. For example, it was shown by Nam et al. (2000); Bernard (2000); Boffetta et al. (2002) in the direct enstrophy cascade that as the friction coefficient ↵ increases, so does the enstrophy spectrum slope, deviating more and more from the Kraichnan prediction, i.e. E(k) k−(3+⇠) where ⇠ is related to the / distribution of finite time Lyapunov exponents (see below). A consequence of (2.15) and ! ! (2.16) is that for ↵ = 0, ⇠ = 0 and for 0 < ⇠ < 2, ⇠ = ⇣2 , where ⇣ is the vorticity structure function exponent. The slope steeper than k−3 when ↵ > 0 implies the differentiability of the velocity field (see section 2.2.2) and the logarithmic correction in (2.25) and (2.26) is absent. In the inverse energy cascade range, it may be shown that the effect is the reverse: as ↵ decreases, energy builds up into a large scale condensate. The apparent effect is a steepening of its slope. This tendency is illustrated for various spectra coming from 2 2 the simulations with resolution Nx = 4096 performed in the framework of the study at chapter 3. The value of ↵ was selected in such a way that the accumulation of energy is prevented at large scales, without depleting too much the inertial range cascade in order 20 CHAPTER 2. DEFINITIONS AND CONCEPTS

10 α 9 =0.156 α =0.228 8 α =0.264 7 3

/ 6 5 k ) k ( 5 E

4

3 101 102 103 Wavenumber k

Figure 2.3: Energy spectra for various Ekman friction coefficients ↵ in the two dimensional inverse energy cascade with k = 103. The spectra are compensated by k 5/3. Dashed- f k k lines represent fit in the inertial-range. Contrary to the direct cascade, the slope diminishes as the friction coefficient increases. to get a spectral slope as close as possible to k−5/3.

Intermittency in 2D turbulence

While the velocity field in three dimensions is known to be intermittent, the direct and inverse cascade deserve separate discussion in the two-dimensional case. In the inverse energy cascade, dimensional scaling was observed in numerical simulations by Boffetta et al. (2000) for the Lagrangian structure function of order up to p = 7, ruling out the possibility of intermittency similar to that in the 3D case. They nevertheless observed an antisymmetric part for high fluctuations of the longitudinal velocity differences, so that there is no Gaussianity. Other experimental and numerical works lead to the same conclusion (Paret & Tabeling, 1998; Chen et al., 2006b; Xiao et al., 2009). In the direct cascade, velocity doesn’t display any intermittency, so that its incre- ment are Gaussian even at small scales. Rather, it is the vorticity structure function that displays anomalous scaling. The exponents ⇣(p) of Sp may be related to the following dynamical argument. As stated in section 2.1, the Lyapunov exponent λ is obtained in the limit when two initially close trajectories in chaotic flows have diverged during an in- finite time. When this time t is finite, these exponents depend on initial separations and are characterised, owing to the large deviation principle, by a probability density func- 2.2. TURBULENCE IN TWO DIMENSIONS 21 tion P (λ t)= tG00(λ)/2⇡ e−tG(λ) (Ott, 2002). The convex Kramer function G(λ) is then | related to the exponents by the relation ⇣2n = min[2q, (G(λ)+2q↵)/λ](Neufeld et al., p h 2000). This implies that the probability density function of the vorticity increment is not self-similar and deviates from Gaussian at small scales (Tsang et al., 2005). A third sign of intermittency is the multifractal property of the vorticity dissipation field, r!(x) 2. One way to get a measure of a chaotic attractor in a give phase space is k k to look at its Renyi dimension spectrum Dq. Dividing the phase space in (hyper)-cubes Cj of size ✏, is defined by (Renyi, 1970): Dq q 1 log j µ(Cj) q = lim . (2.28) D ✏!0 1 q ⇣ln(1/✏) ⌘ − P µ is the natural measure associated with the attractor such that µ(C )=1. is a non- j j Dq increasing function of q and is independent of q for non-fractal attractors. In particular, P is the information dimension and the correlation dimension. These dimensions D1 D2 may be determined numerically using box-counting algorithms (see section 3.3.2 for an example of the measure of ). Tsang et al. (2005) showed that anomalous scaling of D2 the vorticity structure function yields multifractality of vorticity dissipation through the ⇣2q−q⇣2 relation Dq =2+ q−1 .

2.2.3 Energy and enstrophy budgets One can derive the equations for the evolution of the fluid energy and enstrophy by mul- tiplying Navier-Stokes equations (2.18) respectively by the fluid velocity u and vorticity !. Only the energy conservation terms are written, the case of enstrophy being analogous. The equation for the instantaneous variations of the shell-averaged energy content at wave numbers such that k = k reads: k k @ E(k)+Π(k)= 2⌫Z(k)+ ↵E(k)+F (k). (2.29) t − − Π(k)= u (u ru) is the non-linear transfer contribution which satisfies h · · i 1 Π(k)dk =0. (2.30) Z0 This term also corresponds to the integral of the triadic interactions over the wave-numbers k = k. This quantity, represented along the k axis, allows one to better visualise the k k < k cascades. Indeed, representing Π (k)= 0 Π(k)dk on a plot with a logarithmic scale in the wavenumber dimension displays a plateau, i.e. a constant flux. R For example, in the inverse energy cascade, the energy goes from low to large values of k.Π<(k) is thus a source term for the large scales (larger than the forcing scale). Hence, as this term appears on the right hand side with a negative sign, it will give a negative 22 CHAPTER 2. DEFINITIONS AND CONCEPTS

Π 0.02 (k) −αE(k) −2νE(k) 0.01

0

-0.01

-0.02

100 102 Wavenumber k

Figure 2.4: Illustration of the terms appearing in the spectral energy budget (2.29) in the 2 2 inverse cascade at a resolution Nx = 4096 . The peak in the non-linear transfer is due to the stochastic forcing injected at the corresponding scale. plateau. In the direct cascade, this term would be constant and positive for the enstrophy budget, as this quantity cascades to the small scales. The energy dissipation has two contributions, which are on average negative for all k. 1 One comes from molecular viscosity with total dissipation ✏ = 0 2⌫Z(k)dk, the other one from Ekman friction with total dissipation ✏ = 1 ↵E(k)dk. The term F (k)= ↵ 0 R uˆ(k) f(k) denotes the input power at scales k, where the average is taken over wave- h · i R numbers with modulus k. Figure 2.4 illustrates the non-linear term along with the friction term appearing in equation (2.29). Those simple relations may serve as a benchmark when designing Navier- Stokes solvers. In a statistically steady state, @ E(k) is zero, and the other terms must, h t i on average, balance.

2.3 Relative dispersion rates of Lagrangian trajectories

2.3.1 Lyapunov exponents One important quantity to characterise the dynamics of transported elements in chaotic flows, coming back from the works of Lyapunoff (1907), describes how two tracers in- finitesimally close do separate asymptotically in time following the continuous stretching, contractions and rotations of their separation vector. Tracers are particles that follow ex- actly fluid stream lines and can be thought as attached to fluid elements, hence their name 2.3. RELATIVE DISPERSION RATES OF LAGRANGIAN TRAJECTORIES 23

(see 3.1.1). They are characterised by their position X, and velocity V = u(X), which is the fluid velocity at their position. Their equation of motion reads:

X˙ = u(X,t). (2.31)

Considering the tangent bundle in the phase-space R(t)=δX(t) at point X(t), its evolution reads: dR(t) = σ(t)R(t), (2.32) dt with σij = @jui(X(t)) denoting the Lagrangian strain matrix. The integral

t J = exp0 σ(s)ds , (2.33) ✓Z0 ◆ with expo the time-ordered exponential, is the Jacobian matrix such that R(t)=JR(0). Consider an initial volume of fluid which is evolved by the dynamic. It will be elongated along some directions and stretched along others. Actually, for each single trajectory and in the limit t , the orientation of this ellipsoid’s axis will have converged (due to !1 the Multiplicative Ergodic Theorem by Oseledec (Oseledec, 1968)) in the directions of the T 2 T eigenvectors ej of the matrix J J. Indeed, with R(t)=JR0 then R = R R = T T k k R0J JR0. J J is a symmetric matrix, hence diagonalisable. Because it is also positive, its eigenvalues are positive and may written in the form of an exponential eλj t, defining the Lyapunov exponents: 1 λj = lim ln ( Jej ) j =1,...,2d. (2.34) t!1 t | | 2d The evolution rate of a phase-space volume is given by ✓ = j=1 λj. In incompressible flows, this sum is zero, and the volumes are thus conserved. To provide another example, P as will be discussed in 4, for inertial particles, whose dynamic is dissipative, this sum is negative, yielding a contraction rate with a given characteristic time which is a property of the particles. These Lyapunov exponents may be used to define a fractal dimension of a phase-space attractor, called the Lyapunov dimension. Ordering the exponents in decreasing order i and defining the partial sum S(i)= j λj, then dL is the interpolated index for which S(d ) = 0. The Kaplan–Yorke conjecture (Kaplan & Yorke, 1979; Eckmann & Ruelle, L P 1985) then states that the information dimension of the attractor D1 is equal to dL.

2.3.2 Separation rates The velocity of the relative separation between two tracers is not a trivial quantity in turbulence. Denoting by R(t)=Xi(t) Xj(t) the separation between two particles i and j at time − t, the question is to know how fast such particles will migrate away from each other. It 24 CHAPTER 2. DEFINITIONS AND CONCEPTS may depend on multiple factors like the initial and present separations R(0),R(t) and the statistics of the fluid velocity differences projected on their separation vector

R δ u =( u(x) u(x + R) ) . (2.35) R k k−k k · R k k

Smooth flows

Consider first two particles separated by a distance below the dissipation scale ld, i.e. in the dissipative range. The flow is then differentiable and Lipschitz in R, i.e. δ u σR, R ⇠ ensuring the unicity of the solution of (2.31). The particles then separate exponentially in time with the Lyapunov exponent λ:

R(t) = R eλt. (2.36) k k k 0k

Non-smooth flows Consider now two particles initially separated by R which is in the inertial sub-range. k 0k In this range, in the 3D direct cascade or 2D inverse cascade, velocity field is no longer smooth: δ u r1/3. This implies dR2/dt =2R δ u R4/3 and: h r i/ · R / 2 3 R(t) gRt , (2.37) k k R0 / D E where gR is the Richardson constant and the average is taken over particle pairs initially at distance R . This rate is faster than diffusion (R2(t) t) and the regime is called 0 / super-diffusive, faster than what can be attributed to sole chaotic motions. Indeed, in the exponential separation (see paragraph above), the time for two particles to reach a scale R diverges logarithmically with R/R0. On the contrary, the explosive separation does not depend on the initial separation R0 and particles will always reach R in a finite time. This observation was already predicted by Richardson (1926). He measured a scale- dependant diffusivity K which fits well with K(r) r4/3 on 4 decades. Indeed, a contam- / inant cloud of size r is only advected by vortices larger than r, and its diffusions results mainly from vortices of size r. From this result, he derived equation (2.37) using Fickian dif- fusion. This scaling law for K(r) was later formulated in the framework of the Kolmogorov theory following the Obukhov hypothesis (Obukhov, 1941). If r is in the inertial range, the effective diffusivity K(r) must only depend on r and ✏, leading to K(r) ✏1/3r4/3. / This super-diffusive behaviour however was showed by (Batchelor, 1950) to be preceded by a ballistic separation, i.e. with a velocity constant in time, for which we get R(t) 2 = k k 2/3 2 C(✏R0) t . This regime is valid during the correlation time of the eddies ofD size R0. ThisE correlation time is typically of the order of the eddy turn-over time associated with the scale R, ⌧ ✏−1/3R2/3. From this observation, Bourgoin (2015) and Thalabard et al. R / 2.3. RELATIVE DISPERSION RATES OF LAGRANGIAN TRAJECTORIES 25

(2014) successfully proposed to interpret the explosive separation as an iterative ballistic process. This Richardson dispersion regime is also associated with another issue about the re- versibility of pair separations: in three dimensions, the Richardson constant gR is not the same when considering the forward in time evolution of pairs and its backward in time equivalent. Heuristically, this can be understood by the fact that the odd number of di- mension allows a fluid ellipsoid to be elongated along more dimensions than those along which it can be squeezed. This would not be the case in two dimensions. 26 CHAPTER 2. DEFINITIONS AND CONCEPTS Part I

Turbulent dispersion and mixing

27

CHAPTER 3

Tracers dispersion in two dimensional turbulence

Contents 3.1 Introduction ...... 29 3.1.1 Diffusion at long times ...... 33 3.1.2 Continuous source ...... 36 3.2 middling version ...... 36 3.2.1 Fluid phase integration ...... 36 3.2.2 Injection mechanism ...... 38 3.2.3 Removal mechanism ...... 39 3.3 Results ...... 39 3.3.1 One point dispersion ...... 39 3.3.2 Two-point correlation ...... 43 3.3.3 Phenomenological description ...... 49 3.4 Brief conclusion ...... 54

3.1 Introduction

Turbulent mixing is of particular concern in situations such as the formation of clouds through condensation of small water droplets (Grabowski & Wang, 2013), gas accretion in planet formation (Johansen et al., 2007) or phytoplankton and nutrients distribution in the oceans (Mann & Lazier, 2013). Mixing refers to the evolution of an initial distribution of a scalar field (temperature, salinity, or the concentration of any substance...) by the fluid.

29 30 CHAPTER 3. TRACERS DISPERSION

The mechanical stress induced by the stirred flow tends to deform the initial distribution of this field, and the multi-scale nature of turbulent flows gives rise to very complex shapes and patterns of the concentration field (Celani et al., 2001). For example, while the scalar is also submitted to molecular diffusion, which tends to smooth out concentration gradi- ents, mixing by stretching and compression in directions orthogonal to each other cause to reinforce these gradients by creating elongated concentration filaments. Because incompressible flows preserve volumes, an homogeneous initial scalar concen- tration remains uniform at any later time. However, non-uniform initial patches of concen- trations will be deformed by the swirling eddies and create locally high gradients. Figure 3.1 displays an instantaneous field of scalar concentration advected by a turbulent flow: large fronts and cliffs are seen along with rather uniform regions. These gradients form because turbulence brings close together trajectories of fluid elements carrying different scalar tra- jectories and history. Scalar differences over small scales grow in intensity while the front boundaries become thinner, until they are eventually dissipated by molecular viscosity. These large differences are responsible for strongly intermittent statistics in the scalar dis- tribution (Sreenivasan & Antonia, 1997), i.e. the probability density function (pdf) of scalar value shows a departure from a Gaussian behaviour, and displays exponential tails (Pumir et al., 1991) that result from rare, extreme events. They prove that a restrictive vision considering a large number of small-scales, uncorrelated stretching events for the scalar distribution, which would yield Gaussian pdf through the central-limit theorem, is not correct. Interestingly, the passive scalar is strongly intermittent both in 2D and 3D even in the absence of intermittency in the velocity field itself in 2D, and also in simple random Gaussian velocity fields (Shraiman & Siggia, 2000). Similarly to the velocity increments in 3D or the vorticity in the 2D direct cascade, the scaling exponents of the scalar structure function Sn(r)= (✓(x) ✓(x + r))n r⇣(n) are not linear: ⇣(n) = n/3 (see section ✓ h − i/ 6 2.1.1). Scale invariance is thus broken and Sn reads (Celani & Vergassola, 2001):

⇣dim(n)−⇣(n) dim L Sn(r) r⇣ (n) (3.1) ✓ / r ✓ ◆ The difference ⇣dim(n) ⇣(n) is the correction for the anomalous scaling, and the broken − scale invariance manifests in the presence of L although r L. ⌧ Some analytical models for the velocity correlations, like the Kraichnan model (Kraich- nan, 1994), allow to recover predictions about the behaviour of limn!1 ⇣(n). In the Kraich- nan ensemble, the two-points, two-times velocity correlation are:

v (x ,t )v (x ,t ) = D (x x )δ(t t ) (3.2) h i 1 2 j 2 2 i ij i − j 1 − 2 with r r D (r)=D δ D r⇠[(d + ⇠ 1)δ ⇠ i j ] (3.3) ij 0 ij − 1 − ij − r2 3.1. INTRODUCTION 31

for r smaller than the integral scale. ⇠ denotes the degree of roughness of the flow: the velocity field smoothness increases with ⇠ and is differentiable for ⇠ = 2. In this model, and under the additional assumption of high dimensionality, d ⇣(2), � it was analytically shown in Balkovsky & Lebedev (1998) that there exists a critical order n such that n>n, ⇣(n) is independent of n This asymptotic behaviour seems also to c 8 c be observed with direct numerical simulations of two dimensional inverse cascade, where it was estimated in Celani et al. (2000). As vorticity and passive scalar share the same transport equation, it is tempting to com- pare their scaling laws to see if the exhibit similarities. However, the direct link between ! and u make the equation for ! non-linear, which can lead to discrepancies for small scale quantities. For example, Dubos & Babiano (2003) have shown using numerical simulations that this difference is responsible for faster temporal fluctuations of the vorticity gradients. In Boffetta et al. (2002), a correspondence is made between the intermittency of vorticity and that of a passive scalar transported by the flow, showing that ⇣! = ⇣✓ p. This corre- p p 8 spondence may be explained using the following ad-hoc argument (Tsang et al., 2005) based 1/2 on the Lyapunov exponent λ (see section 2.3). Since λ u 2 1 k2E(k)dk ⇠ kr k ⇠ kf and assuming, then λ k−⇠/2. Thus λ (and u) characterisingD smallE separationsqR stretch- ⇠ f r ing, are determined by large scale structures, and the small scale vorticity components behave like scalar advected by the large scale flow. Structure functions of order n are linked to the equal time n-point correlation function of the scalar field. For example, consider the following equality for n = 2:

S (r, t)= (✓(x + r,t) ✓(x,t))2 (3.4) 2 − = D✓(x)2 + ✓(x + r)2 E2 ✓(x + r,t)✓(x,t) (3.5) − h i =2(C (0,t) C (r, t)) . (3.6) ⌦ 2 ↵ −⌦ 2 ↵ The last equality results from homogeneity and isotropy and C (r, t)= ✓(x + r,t)✓(x,t) . 2 h i Averages are taken over the positions x. The generalisation of this quantity to n-points displays the link with the n-point joint transition probability for the Lagrangian motion. For a set of n particles initially at position x0,...,xn at instant t0:

C (x ,...,x ; t)= ✓(x ,t),...,✓(x ,t) (3.7) n 1 n h 1 2 i t = ✓(x0,t),...,✓(x0,t ) p (x ,...,x ,tx0,...,x0 ,t )dx0 ...dx0 1 2 0 n 1 n | 1 n 0 1 n tZ0 (3.8)

where p(...) expresses the joint probability that the n trajectories initially at positions 0 0 x1 ...,xn are transported at x1,...,xn at time t. 32 CHAPTER 3. TRACERS DISPERSION

The quantity Cn can be related to the joint motion of n particles. It has a geometrical interpretation in terms of Lagrangian trajectories. For example, in Celani & Vergassola (2001), the intermittency of the passive scalar advection is attributed to long lasting clus- tering of n-tuple of particles. In Bianchi et al. (2016), the shape of spherical puffs of particles emitted is monitored as a function of time, showing that although the puff is initially spherical, the quick and strong distortions prevent the cloud to return back to a spherical shape at later times. It is also shown not to affect much large scale transport statistics, like the pdf of durations of hits and between hits of a downstream target.

In particular, the two-point scalar correlation allows one to express pair dispersion statistics. This correspondence was for example used in Boffetta & Celani (2000) to link frequent pairs encounter and scalar fronts formation. This object, C2(r) may be analytically derived only under drastic constrain on the flow, like for example the Kraichnan ensemble, In such a flow, Celani et al. (2007) have studied scaling properties of a scalar continuously emitted from a point source and derived an exact relation for the two-points equal-time scalar correlation function C(x , x ,t)= ✓(x ,t)✓(x ,t) , demonstrating the persistence 1 2 h 1 2 i of inhomogeneities at small scales.

Figure 3.1: Illustration of a scalar field mixed by turbulent flow, representing a 2D slice 3 3 from a 3D DNS simulation at Nx = 4096 with a mean gradient scalar source. When initial inhomogeneities or inhomogeneous scalar sources are present, mixing by eddies create fronts where the scalar variations over very small scales are of the same order than the rms value itself. 3.1. INTRODUCTION 33

3.1.1 Diffusion at long times In this section, the correspondence between long time displacements of discrete particles and the scalar diffusive behaviour is explicited. Tracers are particles solely advected by the flow, and thus that perfectly follow stream- lines. Denoting their position by X and velocity V , their equation of motion simply reads: X˙ (t)=V (t), (3.9) V˙ (t)=u(X,t), (3.10) where u(X,t) is the flow velocity at time t and at the particle position X. Starting from position X(0) at t = t0, defining its displacement by : t X(t) X(0) = u(X(⌧), ⌧)d⌧. (3.11) − Zt0 The mean square displacement then reads: t t X(t) X(0) 2 = u(X(⌧), ⌧) u(X(⌧ 0), ⌧ 0) d⌧ d⌧ 0. (3.12) k − k 0 0 · D E Z Z In turbulent flows, the exact behaviour of⌦ the integrand is of course↵ unknown. But there are two asymptotics in which we can approximate this integral. The first one, trivial, is when considering very small durations t t , such that u − 0 may be considered constant, or self-correlated during this time interval, leading to X(t)= X + u(t )(t t ), so that 0 0 − 0 X(t) X(0) 2 = u2 (t t )2. (3.13) k − k rms − 0 2 2 2 D E where urms = ux + uy is the mean square velocity averaged over the two dimensions x and y. ⌦ ↵ The second exploits the fact that the autocorrelation of the tracers velocity u decreases to 0 over a finite time scale TCL , the Lagrangian correlation time, defined as: 1 1 X X TCL = 2 ui( (0), 0)ui( (⌧), ⌧) d⌧. (3.14) urms 0 h i Xi Z In the limit of large times (t T ), (3.12) becomes: � CL t 2 2 2 X(t) X(0) TCL urms d⌧ TCL urmst. (3.15) k − k ⇠ 0 ⇠ D E Z This result may also be formulated the following way: decomposing the interval [t0,t] into a subset of N intervals of length TCL ,[t0+iTCL ,t0+(i + 1)TCL ]. N t0+(i+1)TCL X(t) X(0) = u(⌧)d⌧ (3.16) − t0+iTC Xi=0 Z L 34 CHAPTER 3. TRACERS DISPERSION

This long-time displacement is thus a sum of random variables identically distributed, hence, invoking the central limit theorem, it is itself a random Gaussian variable with variance: X X(0) 2 =2u2 T t. (3.17) k − k rms CL Beside the Lagrangian descriptionD of particleE dispersion, a continuous concentration field ✓(x) can be defined, which is also transported along the fluid trajectories. Let us define this scalar quantity in terms of emitted tracers from a source situated at position xS which emits particles at a constant rate φS(⌧)=φS starting from the instant ⇤ t0. The expression of n (x,t) counting the number of particles reaching x at time t then reads:

t n⇤(x,t)= φ δ(x(⌧; x,t) x )d⌧ (3.18) S − S Zt0 where x(⌧; x,t) is the position of the particle at time ⌧ given that it is at position x at time t. φS is an arbitrary number of particles injected at xS at time ⌧. The ensemble average of n⇤, i.e. over a large number of realisations, defines a number density field: t n(x,t)= n⇤(x,t) = φ p(x , ⌧ x,t)d⌧ (3.19) h i S S | Zt0 where p(x , ⌧ x,t) is the transition probability to travel from x at time ⌧ to x at time S | S t. Notice that with such a definition, n⇤ and n both have units of number of particles per unit area, or L−2. This is a consequence of the function δ(x) having units of L−2. In the following, the field n(x,t) is sometimes used in place of the scalar concentration ✓(x,t). ✓ having units of a mass density, the two are simply related by ✓(x)=mpn(x) with mp the constant mass of one particle. As a remark, let us stress that formally, the following distinction has to be kept in mind. The scalar quantity ✓(t, x) is a transported quantity along a Lagrangian trajectory between x0 at t0 and x at time t:

✓(x,t)=✓(t , X(t ; t, x)) = δ(x x(t ; t, x))✓(t , x )dx , (3.20) 0 0 0 − 0 0 0 0 Z while the field n(x,t) results from the forward Lagrangian flow:

n(x,t)= δ(x X(t; t , x ))n(t , x )dx . (3.21) − 0 0 0 0 0 Z Equation (3.17) describes a diffusive process for the particle displacements. The link between ✓ and Lagrangian trajectories (3.20) suggests that the scalar obeys at long times the differential equation: @ ✓ =  2✓ + S(x), (3.22) t T r 3.1. INTRODUCTION 35

2 which is the scalar pendent of equation (3.17). The operator T stands for the turbulent ff 2 r di usion with T = TLurms. S(x) is the source term. In the case we considered, it is a punctual source of constant emission rate φ at position x . Hence S(x)=φ δ(x x ). S S S − S Once again, equations (3.22) and (3.17) are only valid when considering long times t t − 0 since emission or long distances R from the source, i.e. R>urmsTCL , where TL is the time during which Lagrangian velocities are correlated, or equivalently (t t ) > R/u . − 0 rms The advection-diffusion of the passive scalar can be obtained by expressing the conser- vation of ✓ in an elementary volume and reads:

@✓ + u ✓ =  2✓, (3.23) @t · r r with  the molecular diffusion coefficient and u the convection velocity. The link between (3.23)and(3.22) deserves a comment. Decomposing u and ✓ into a mean and fluctuating part, i.e. ✓(x,t)=✓(x)+✓(x,t)0 and u = u(x)+u(x,t)0, we may then substitute in (3.23) and taking the average. Given that u0 = 0 and ✓0 = 0, one obtains: h i h i @ ✓ + u ✓ =  2✓ r u0✓0. (3.24) t · r r − · The last term on the right-hand side denotes the turbulent mixing. To close this equation, the approximation u0✓0 =  ✓ is made, with  > , yielding (3.22) (omitting the − T r T source). As an example, consider atmospheric dispersion. Molecular diffusivity of carbon dioxide in the air is 16 10−6m2s−1 while turbulent diffusion is often estimated as ⇠ ⇥ 30m2/s, hence several orders of magnitude greater. ⇠ The Green function with u = 0 of (3.23) is a d-dimensional Gaussian whose variance decreases linearly with time:

M(t) 1 ✓(x,t)= exp xT Σ(t)−1x . (3.25) (2⇡ Σ(t) )d/2 −2 | | ✓ ◆ M(t)= ✓(x,t)dx is the total mass and is a quantity constant in time, hence preserved by diffusion. Σ is the covariance tensor. Note that the solution for a diffusion process in R a medium moving with constant velocity U is straightforwardly obtained by applying the Galilean transformation x0 = x Ut. − If the diffusion is independent along each direction, the tensor Σ is diagonal. Further- more, if the diffusion is isotropic, the solution reads

A r2 ✓(r, t)= exp (3.26) 2 d/2 −2σ(t)2 (2⇡σ(t) ) ✓ ◆ and only depends on time and distance r from the mean. 36 CHAPTER 3. TRACERS DISPERSION

3.1.2 Continuous source Consider now a continuous emission of particles that takes place during a duration t. Due to the linearity of equation (3.23), the solution can be expressed as the superposition of an 0 0 infinite number of emitting sources from t to t + dt, each emitting a quantity φSdt. The solution then reads:

1 t 1 r2 ✓(x,t)= exp dt0 (3.27) (2⇡)d/2 (t t0)d/2 −4(t t0) Z0 − ✓ − ◆ The value of this integral converges on [0, ] only for d>2. For d = 2, it diverges 1 logarithmically. Why is that so? Actually, diffusion at long times translates into Wiener process trajectories followed by the particles. And it is known that the Wiener process is a recurrent stochastic process in dimensions less or equal than two (Van Kampen, 1992).

3.2 Concentration and mass fluctuations of particles emitted by a continuous point source

The large scalar temporal correlations play a role in what is called vortex trapping. Indeed, particles trapped inside a long-lived coherent vortex allows for interactions which take place at a distance of the size of the eddy to last longer and affect both the suspended solid phase as well as the carrier flow. This phenomenon is for example currently believed to play a key role in planetesimal accretions. Indeed, in planet formation, one challenging step is the understanding of the formation of planetesimals of the kilometre size, and the existence of long-living vortices in protoplanetary disks capable to concentrate large dust concentrations constitute a promising theory (Meheut et al., 2012). The challenge of the system that is studied in this chapter, i.e. the continuous emission from a source, lies in the fact that both spatial and temporal correlations play a role in the n-point concentrations. The addition of the time difference in the correlations adds a non trivial complexity. In this chapter, the problem of continuous mass release is addressed. Having introduced to the reader the dispersion dynamics of both Lagrangian tracers and scalar, the chosen framework to study the dispersion from a continuous point source in two-dimensional turbulence is now described.

3.2.1 Fluid phase integration The flow regime we considered is the inverse turbulent cascade. Direct numerical sim- ulations have been performed using pseudo-spectral (Fourier) scheme in a d-dimensional periodic domain. The flow is forced at high wave-numbers. To maximize the inertial range and to minimise the range of scales affected by viscosity, we chose to implement hyper- viscosity, which translates into a higher power of the Laplacian p>2 in the Navier-Stokes 3.2. CONCENTRATION FLUCTUATIONS 37

(a) t =0.22 TL (b) t =1.12 TL

(c) t =4.5 TL (d) t = 33.7 TL

Figure 3.2: Illustration of recurrence phenomenon. In a few integral times TL, particle distribution becomes nearly uniform in the domain due to large amount of returns near the source. The cut-off distance is Rmax =2L and width of the window is L. equations. In all the simulations presented here, we have chosen p = 8. An illustration of 2 2 the kinetic energy spectrum for a resolution Nx = 4096 is displayed in Figure 3.3. In the two-dimensional inverse cascade, this number is defined as the ratio between the friction and forcing scales:

l 2/3 k 2/3 Re = ↵ = f , (3.28) ↵ l k ✓ f ◆ ✓ ↵ ◆ 1/2 −3/2 where l↵ is the friction scale defined by l↵ = ✏ ↵ , and ✏ is the intensity of the nonlinear energy flux going from small to large scales. More details about the numerical integration of two-dimensional turbulence may be found in appendix A.1. 38 CHAPTER 3. TRACERS DISPERSION

10-1

10-2

10-3 ) k ( E 10-4

10-5

10-6 100 101 102 103 Wavenumber k

Figure 3.3: Fluid kinetic energy spectrum for the two-dimensional inverse cascade with 2 2 hyper-viscosity. The resolution is Nx = 4096 and the flow is forced at k = 1000 with a white noise. The black dashed line corresponds to the dimensional scaling k−5/3.

3.2.2 Injection mechanism

Numerically, a number of particles φS is injected into the domain for a period t = 1. The symbol φS thus stands for the injection rate, in units of number of particle per unit time −1 3 [T ] , or equivalently mass per unit time, [M]/[T ], through the relation m = ⇢pa φS, with ⇢p = ⇢f the particle density and a the particle radius, considered as infinitesimal. During a time step ∆t, φS∆t particles are thus emitted. Two different mechanisms were considered for the injection of particles from the source. The first one consists of a uniform seeding around the source, either in a square region with sides Rx and Ry, or a circular region of radius R. This may be used to emit regular puffs of particles at a given period Tinj and is used for example in Scatamacchia et al. (2012) to study strong deviations from Richardson separation law, or in Celani et al. (2014) to determine the probabilities to detect concentrations above a given threshold downstream of the emission. Another way of releasing particles into the domain is proposed, the reason why will be explained in section 3.3.3. Initially, at the beginning of the simulation, a first particle is randomly seeded close to the source. Its position and identifier is noted X0. Then, φS ∆t particles are injected at positions which are linearly interpolated between the source and X0:

(xS X0) Xi = X0 + i − i =1,...,φS∆t. (3.29) φS∆t

During the next time step, all particles will have moved further, and X0 takes the value of the position of the last emitted one which becomes the new reference for the next time 3.3. RESULTS 39 step. This mechanism, referred later as line injection, requires the period Tinj to be short enough so that X x is not too large compared to small scale structures, otherwise the k 0 − Sk injection creates unrealistically aligned structures traversing (and not affected by) eddies. Hence, it should be at most comparable to the dissipation time scale. In our case, the chosen period is equal to the time step (Tinj = ∆t) which is the best possible choice to respect the approximation of a continuous source.

3.2.3 Removal mechanism

Boundary conditions must be implemented to remove particles from the system in order to reach a statistical steady state for the total number of particles. Furthermore, if one wants to achieve reasonable statistics to measure, for example, fractal dimensions at a given distance R from the source, both a sufficient number of particles need to be present and during a time long enough to be able to perform ensemble averages as large as possible. This may quickly leads to unmanageable number of particles present in the domain and pauses difficulties for both computation capacity and memory. Hence, absorbing boundary condition are used: as soon as a particle goes beyond a given criterion it is removed from the domain. In our experiments, depending on the quantity of interest, two different choices were implemented. The first one is a spatial criterion, allowing particles to live only inside a given boundary Γ. In all our simulations, this frontier consists of a circle centered around the source with a fixed radius Rmax. The second considered boundary condition is a temporal one, allowing particles to live only until a maximum age tpmax . Additional details regarding the used software, LAGSRC2D, may be found in ap- pendix A.2.

3.3 Results

3.3.1 One point dispersion

Mean square displacements and Lagrangian velocities autocorrelation

The mean square displacement of particles is defined as:

R(⌧)2 = X (t) X (t ⌧) 2 . (3.30) k p − p − k ⌦ ↵ D E For small ⌧, as long as the Lagrangian velocity of the tracers is correlated, these are expected to migrate from their initial position at a constant speed, in average, yielding 2 2 2 R(t) = urmst . For times longer than the Lagrangian velocity correlation time TCL , the diffusive regime is expected to be recovered with R(t)2 t. ⌦ ↵ / T is defined in the following way. The Lagrangian velocity autocorrelation functions CL ⌦ ↵ 40 CHAPTER 3. TRACERS DISPERSION are given by: V (0) V (⌧) C(⌧)=h p · p i. (3.31) V 2 k pk In statistically stationary flows, this quantityD is independentE of time and depends only on the time increment ⌧. It is well known that this quantity decays exponentially with time as it was already conjectured in Taylor (1921). From this function, the Lagrangian correlation time reads: 1

TCL = C(⌧)d⌧. (3.32) Z0 When ⌧ is small, the correlation C(⌧) is constant and does not move appreciably away from unity. The velocity is differentiable with respect to ⌧ and C(⌧) may be developed in power series around ⌧ = 0. To the leading order, one has:

t X (t) X (0) 2 =2 (t ⌧)C(⌧)d⌧ = u2 t2. (3.33) k p − p k − rms D E Z0 and the separation between the two particles is said to be ballistic, i.e. taking place at a constant rate. Notice that the mean square displacement is expected to be isotropic because the forc- ing of the flow is itself isotropic. By measuring and comparing the displacements along the x direction R2(t) and along the y direction R2(t) (not shown here), these were − x − y observed to be identical t. This equality does not fully ensures isotropy but strongly ⌦ ↵ 8 ⌦ ↵ suggests it. In order to evaluate (3.30) and (3.31) in the case of continuous particle emission in the 2 2 inverse energy cascade, a numerical simulation is run at a spatial resolution Nx = 512 , with small-scale, delta-correlated forcing at k [100, 105]. A fixed number of 5 10−6 tracers f 2 ⇥ are uniformly spread in the square domain of size L2 = (2⇡)2. No boundary condition to remove the particles are applied. Figure 3.4 (left) displays C(⌧) for various values of the Reynolds numbers Re↵. One can see that C(⌧) is indeed a constant function with value unity at small times, falling to zero after a time comparable to the large eddies turn over time T↵ = l↵/urms. Figure 3.4 (right) reports the mean square displacement R2(t) as a function of time and for various Reynolds numbers, which was adjusted by varying the Ekman friction ⌦ ↵ coefficient ↵. The ballistic regime is clearly seen at small times, and the agreement with the 2 2 ff dotted-dashed line representing urmst is excellent. The di usive regime is also represented by the dashed line. The lowest Reynolds used was actually chosen to be below the turbulent transition. It is nevertheless represented on the figures to stress that the necessary ingredient to obtain this succession of regimes - ballistic then diffusive - is only to have a finite Lagrangian correlation time, whatever the turbulent state of the flow. 3.3. RESULTS 41

1 Reα = 67.3 Reα = 61.93 -2 Reα = 54.1 10 α

⟩ Re = 42.94 ) 2 α τ Reα =3.408

/l -4 + ⟩ 10 2 t ) ( t u ( ) t R ( ⟨ -6 u 10 ⟨

10-8 0 100 10-4 10-2 100 102 104 τ /Tα t/TCL

Figure 3.4: Left: Lagrangian velocity correlation functions C(⌧) in log-lin plot. Curves are shown to superimpose when represented as a function of ⌧/T↵ with T↵ = l↵/urms. Right: mean square displacements as a function of time for the same Reynolds numbers.

TCL is the Lagrangian correlation time. Dotted–dashed line represents the ballistic regime 2 2 2 R = urmst for the highest Reynolds which fits very well with the data. Dashed line represents the diffusive regime with R2 t. ⌦ ↵ / ⌦ ↵ These measures, together with the ones for the equilibrium density probability, confirm the validity of the ballistic and diffusive approximations for the tracer motion injected by a point source in the two-dimensional inverse cascade.

Equilibrium distribution As stated above, when including absorbing boundary conditions, the total number of par- ticles present in the domain reaches a statistically steady state, for which the average of Np over a given time interval T is then N = cst. One can wonder about the equilibrium h piT distribution for the number of particles as a function of the distance from the source, n(R). As explained in the introduction of this chapter (see 3.1), point particle dispersion can be related to a scalar field ✓(x). To obtain a theoretical prediction for this quantity, two regimes must be considered, either if we are close or far from the source. For short travelled distances (or times from ejection), particles mainly see a constant velocity urms. They are thus shot radially from the source and distributed uniformly on an enclosing surface, i.e. a circle in two dimensions. For a perfect continuous emission, this ballistic regime creates a field of concentration φ ✓(R)= S . (3.34) urmsR For large distances, the scalar mixing is expected to be dominated by diffusion as expressed by equation (3.22). Averaged over time, it yields the stationary solution ✓ : h i 42 CHAPTER 3. TRACERS DISPERSION

0= 2 ✓ + φ δ(x x ). (3.35) T r h i S − S Integrating (3.35) on a disk of radius R centered on the source (denoted by ΓR), then using the Stokes theorem and assuming isotropy, we get:

 r ✓ e =  (@ ✓ )Rd−1dΩ = φ , (3.36) T h i · r T R h i d − S ZΓR ZΓR where d is the dimension and dΩd the solid angle element in d dimensions (for d = 2, it is an arc of a circle). Owing to isotropy:

d ✓ φ R1−d h i = S . (3.37) dR − T 2⇡ For d = 2, the solution reads:

φS log R ✓(R) = ✓ 0 . (3.38) h i h i − T 2⇡

Absorption condition is ✓(Rmax) = 0. Then:

φ log (R/R ) ✓(R) = S max . (3.39) h i − 2⇡T A time scale t⇤ at the crossover of these two regimes may be defined, i.e. beyond which the diffusive regime is recovered. t⇤ is associated to a scale R⇤ such that:

t⇤ = R⇤/u =(R⇤)2/ R⇤ =  /u . (3.40) rms T ) T rms To assess the agreement between this expectation in a turbulent inverse cascade, a 2 2 numerical simulation is carried at resolution Nx = 512 in a periodic square domain [0 2⇡] −2 ⇥ [0 2⇡]. Particles are seeded in a uniform region of size l/lf =8 10 surrounding the 2 ⇥ −3 source situated at xS =(⇡, ⇡). 10 particles are injected per time step with ∆t = 10 , 5 hence a rate φS = 10 particles per unit time. The spatial absorbing boundary condition is at Rmax = L/4=⇡/2. Statistical station- arity of the total number of particles in the domain is expected to be reached after a time Rmax/urms. This was indeed verified numerically. The simulation is run for another 900 T , with T = R /u the turn over ⇡ Rmax Rmax max rms time of the eddies of size R . The profile ✓(R) is computed by averaging over time and max h i the number of particles in annulus shells around the source. ✓(R) is displayed in Figure 3.5. The left panel focuses on the density profile close h i to the source and is shown to be indeed decreasing as R−1, in agreement with (3.34) and validating the ballistic approximation for short emission times. The right panel focuses on 3.3. RESULTS 43

3 1 10 0.9

102 0.8 0.7 ⟩ ⟩ 1 0.6

10 N N ⟨ ⟨ / / ⟩ ⟩ 0.5 ) ) R R ( (

0 θ θ 0.4 ⟨

⟨ 10 0.3

10-1 0.2 0.1

10-2 0 10-3 10-2 10-1 100 10-1 100 R/Rmax R/Rmax

Figure 3.5: Average number density ✓(R) as a function of the distance from the emission point, normalised by the average number of particles in the domain N = h i 1 Rmax N(R)dR. Left: focus on ballistic regime at short distances. Red dashed line Rmax 0 corresponds to the solution 3.34. Right: zoom on large R the effect of the absorbing con- R dition solution. Red dashed line is a fit over the large R using the form of the solution (3.39). The parameters of the fit may then be used to estimate the turbulent diffusivity T . the large distances, where the effect of the absorbing boundary condition creates a profile compatible with (3.39), thus ensuring the validity of the diffusion approximation (3.35). We conclude this section by a remark, noting that statistical convergence requires long time averages. Indeed, instantaneous snapshots of the particle distribution are shown in Figure 3.6, each separated by a few integral time scales, revealing strong instantaneous inhomogeneities and anisotropies.

3.3.2 Two-point correlation Figure 3.7 displays an instantaneous distribution of particles in the inverse energy cascade. 2 2 It was obtained in a simulation at resolution Nx = 4096 using the line injection mechanism (see section 3.2.2). Several qualitative comments can be made upon the observed features of the flow.

Starting from the source (indicated by the red cross on the figure) we first note that • the particles migrate away from the source following a nearly straight line. The nearly ballistic regime followed by particles as they separate from the source. They are transported by larger eddies and their trajectory is deformed by small vortices encountered. 44 CHAPTER 3. TRACERS DISPERSION

Figure 3.6: Instantaneous cloud distribution at times 2.5, 5, 7.5, 16 TI . Strong inhomo- geneities due to large eddies transport are observable. The width of the window is L/6. Red cross indicates the source.

Further, strong and sudden deviations from this regime are revealed, where the line of • particles is deformed into a sheet with increasing stretching rate along the transverse direction and compressing rate along the longitudinal direction. This is a manifesta- tion place of homoclinic tangle, where an unstable and a stable manifold mix together. Under these circumstances, it is legitimate to expect fractality in the particle cloud (Manneville, 2010).

Going further away from the source, complex patterns combine coherent lines folded • by the dynamic, still observable at small scales, and a rather uniform background.

The qualitative distribution obviously changes as we go even further away from the • source. At the top of the figure, one can see that the cloud becomes more and more uniform. It is thus expected to recover a homogeneous mass distribution far from the source and at large scales.

Following this qualitative description, we wish to characterise the geometry of the distri- bution and how it varies with the distance R from the source. The complex mix of the differ- ent dynamical regimes and foldings of the line initially emitted gives the intuition of a multi- 3.3. RESULTS 45

Figure 3.7: Zoom on the particle distribution around the source (red cross). The width of the window is L/50 = 20 lf .

fractal distribution (Beigie et al., 1994). One way to measure the fractality of the ensemble is to determine the scaling of the quasi-Lagrangian mass m (R, r)= m (R,r,t) , QL h QL it which relates to the average mass found, on average, around a particle in a ball of size r at a distance R. mQL(R, r) actually corresponds to the two-point correlation

C (R, r)= ✓(X ,t)✓(X ,t) (3.41) 2 h 1 2 it

integrated over a volume r, with R = 1 (X + X ) x and r = X X , hence: 2 1 2 − S k 1 − 2k � � � � r 0 0 mQL(R, r)= C2(R, r )dr . (3.42) Z0

This quantity can also be defined in terms of the two-point transition probability in the following way: let n(x,t) be the number of particles at position x at time t. It is the contribution of all particles emitted in the past t0

Hence: t 1 n(x,t) = δ(x x(t x ,t ))φ δ(x x )dt dx h i − | 0 0 S 0 − S 0 0 −1Z −1Z = φ tδ(x x(t x ,t )) dt S − | 0 0 0 −1Z The integral over the initial position vanishes because all particles originate from the same position xS corresponding to the emitting source. The average product of this quantity at two different positions x and x + y is

n(x,t)n(x + y,t) = φ2 dt dt0 P (x, t, x + y,tt ,t0 ) (3.43) h i S 0 0 | 0 0 ZZ The integrand in this last equation denotes the joint transition probability of finding a particle 1 at position x at time t emitted at time t0 and a particle 2 at position x + y at 0 time t emitted at time t0. It can be decomposed as:

t t n(x,t)n(x + y,t) = φ2 dt P (x,tt ) dt0 P (x + y,tx, t, t ,t0 ) (3.44) h i S 0 | 0 0 | 0 0 −1Z −1Z Integrating over a ball of size r centered on x yields:

dy n(x,t)n(x + y,t) = φ2 dt P (x,tt ) dy dt0 P (x + y,tx, t, t ,t0 ) h i S 0 | 0 0 | 0 0 ZBr Z ZBr Z mQL (3.45) | {z } The under-braced term denotes the quasi-Lagrangian mass and is the object to be evaluated as a function of r. We first chose to compute the correlation dimension , linked to pair probability. Its D2 measure requires evaluating the correlation integral. In practice it is approximated by the discrete count of the number of pairs whose distance is smaller than r:

m = Θ(r X X ) (3.46) QL − | i − j| Xi

This kind of analysis requires a scaling range as large as possible. We thus performed 2 high spatial resolution simulations at Nx = 4096 . In order not to be contaminated by the recurrence problem (see section 3.1.2), we don’t use an absorbing boundary condition at

a given distance from the source but we rather fix a finite lifetime for the particles tpmax . The physical origin of this removal condition is that the concentration transported by fluid elements will gradually fade over time due to diffusion. After a given time, these fluid elements will contribute by only a negligible amount to the quasi-Lagrangian mass. A numerical simulation is run for a duration of 10 T in the statistically steady state ⇠ L for both fluid kinetic energy and particle number. With the criterion of removing particles

after a fixed time tpmax , the number of particles in the domain is exactly constant with

value N = φStpmax . The region around the source is divided into annulus zones at distances R from the source. For each zone, the number of pairs that are separated from a distance below r are counted and averaged in time over 10 large-eddy turn over times. A very large ⇠ number of scanned pairs is considered to reach sufficiently interpretable statistics. It reaches 5.12 1013 and the counting necessitates 5 103 equivalent CPU-hours for each value of ⇥ ⇥ tpmax . For two particles X1 and X2, the probability that their inter-distance is less than r conditioned on the fact that their center of mass is at a distance R from the source reads:

(X + X ) p<(r, R)= X X ∆R, the value of R assigned to each line is not relevant any more. It is only shown for completeness and to illustrate that no clear plateau is observable. Rather, we get a smooth variation of the scaling exponent. This variation actually comes from a contamination from a uniform background that is more and more dominant with increasing R. It has a marked effect on all scales r, as shown by an increase of D(r) at large R even for small r/l 1. The line emitted by the f ⌧ source is thus recovered at smaller and smaller r as R increases. Only close to the source (R & lf ) is the line presence clearly marked. To confront the hypotheses of increasing D(r) because of continuing return from long- living particles, we did measure the same quantities, with the same geometrical framework, and varying the lifetime of the particles tpmax . If particles are allowed to live a shorter time, they contribute less to this uniformisation. One would thus expect the maximum of D(r) to lower. Same quantities are shown in Figure 3.9. It is indeed seen that the maximum of

D(r) decreases with tpmax , and that the line D(r) = 1 is quickly recovered. Only data for an arbitrary R = 52.5lf is shown, but this effect was ensured to be present at all distances R. This validates our starting hypotheses, which makes impossible the estimation of a correlation dimension . D2 It thus appears that the difficulty to determine a unique, clear scaling for mQL results from the combination of a large variety of physical mixing processes: ballistic transport, mixing by turbulent eddies, and uniformisation by long time diffusion. In the next section, another way to determine the scaling of the quasi-Lagrangian mass is proposed.

2 100 R/lf =2.5 R/lf = 97.5 10-2 R ↑ 1.5

-4

10 ) ) r r ( ( 1 R ↑ < 2 D p 10-6

0.5 10-8

-10 0 10 -2 -1 0 1 2 10-5 10-4 10-3 10-2 10-1 100 101 102 103 10 10 10 10 10 r/lf r/lf

< Figure 3.8: Left: p2 (r) for various distances from the source R and tpmax =0.4. Black dashed line is r and red dashed line r5/3. Right: details of the logarithmic slopes D(r) / / for the curves on the left. 3.3. RESULTS 49

2 100

tpmax /TI =0.051

tpmax /TI =0.1

tpmax /TI =0.13 1.5 ) ) r r ( ( 10-5 1 < 2 D p

0.5 tpmax /TI =0.051

tpmax /TI =0.1

tpmax /TI =0.13 -10 0 10 -2 -1 0 1 2 10-5 10-4 10-3 10-2 10-1 100 101 102 103 10 10 10 10 10 r/lf r/lf

Figure 3.9: Same plots as in Figure 3.8 for a fixed value R = 52.5 lf and varying tpmax . TI is the integral time.

3.3.3 Phenomenological description In this section, another approach is presented to describe the quasi-Lagrangian mass fluc- tuations in the emitted particle distribution. Consider an emitted line Γ with a definite length LC . Due to velocity increments fluctu- ations, it will undergo longitudinal stretching and compressing, in addition to bending, and we may define a line density profile ⇢(s, t). The total mass of this line is m = Γ ⇢(s, t)ds(t) and grows linearly in time m(t)=φ t, if no removal term is included. S R We wish to exploit these continuous stretchings and foldings in space to quantify the total mass present in a region of size r at a distance R from the source. Figure 3.10 shows a scheme introducing the notations used hereafter. The idea is the following: a first reference particle P0 at a distance R0 from the source and an age A0 is picked. Among older ones, a particle PH is searched corresponding to the last one to be inside a circle of radius r around P0 before the line escapes further away from P0. In practice, this distance is set to 2r. This limits the very short returns inside the ball, adding a small error r at the determination of the distance R0. The age of PH is noted AH . The line joining particles P0 and PH is represented in light green on the scheme. The average age difference between particles P0 and PH is written as:

∆A(i) = A(i) A(i) (3.49) H0 H − 0 D E D E and is a function of R0, A0 and r. The superscript i denotes the index of the line. The first line just described above has the index i = 0. When this line comes back at a distance below r from particle P , the particle P is marked. It then serves as the new reference  0 R particle and the algorithm is repeated. The new line then has the index i = 1. In the following, this superscript is not written except where needed. 50 CHAPTER 3. TRACERS DISPERSION

(0) r P (1) 0 PH R (0) 0 PH

2r

P (0) = P (1) x R 0

Figure 3.10: Scheme of the setup based on particle ages to determine quasi-Lagrangian mass scaling. The red cross represents the source from which particles are emitted. The contributions of the mass around a reference particle P0 inside the circle of radius r come from the green portions of the line.

The quantity (3.49) is a measure of the (average) mass between particles P0 and PH . Indeed, it corresponds to what is emitted during a time ∆A , hence m = φ ∆A . h H0i S h H0i The quasi-Lagrangian mass may then be viewed as the total contribution from all these lines inside the ball of radius r:

1 m = φ A(i) A(i) . (3.50) QL S H − 0 Xi D E

Theoretical predictions

Multiple regimes can already be guessed using what is known about relative dispersion in turbulent flow (see section 2.3). Let r(t) be the distance between particle P0 and PH at time t and rE the distance between these same particles at the moment tE, when the particle P0 is emitted. rE is thus the distance that the elder particle PH will have travelled from the source before particle P0 is emitted. This separation is of the order of rE = urms∆AH0. Different cases described here below can be distinguished. Notice that the characteristic length scale used to compare rE is the forcing length scale lf . In our two dimensional simulations focusing on the inverse energy cascade, i.e. with a very small scale forcing to maximise the inertial range, this is tantamount to compare rE with the dissipative scale lD. Below lf , the flow is actually differentiable.

Case 1: r l • E ⌧ f 3.3. RESULTS 51

– Case 1a: r(t) l ⌧ f The distance between the two particles always remains in the sub-forcing scales, ⇠t hence in the dissipative regime of the flow, so that r(t)=rEe , where ⇠(t, rE) is the positive finite-time Lyapunov exponent, thus r(t) ∆A . / H0 – Case 1b: r(t) l � f The separation can be divided in two steps. Firstly, initial separation has started in the dissipative regime and lasts until the separation equals the forcing scale: r(t)=l = r exp (λt⇤), hence during a time t⇤ = 1 log lf . Secondly, once f E λ rE their distance is above the forcing scale, separation will be ballistic during a

turn-over time associated with the scale lf ,⌧rE , and will after be dominated by 1/2 ⇤ 3/2 the Richardson explosive regime : r(t)=✏ (A0 t ) . From this relation, we 2/3 − have that r l er(t) −A0 . Hence, for A r2/3✏ 1/3, the quantity ∆A is E / f 0 � − H0 independent of r.

Case 2: r l • E � f −1/3 2/3 Case 2a: A0 < ✏ rE = ⌧rE Particles P0 and PH are initially distant from rE >lf . They separate during a time A0 which is inferior to the eddy turn-over time associated to their initial separation

⌧rE . This corresponds to the ballistic regime prior to Richardson explosive separation, ff where particles see a constant velocity di erence δrE u, so that r(t)=rE + A0δrE u. For A ∆A u /δ u, it is thus also expected to recover ∆A r as in case 0 ⌧ H0 rms rE h H0i/ 1a. −1/3 2/3 Case 2b: A0 > ✏ rE = ⌧rE

If particles have separated during a time A0 longer than ⌧rE , the inter-distance is then in the explosive Richardson regime: r(t) A3/2 and is independent of the / 0 initial separation rE.

Figure 3.11 (left) summarises the relations listed above.

Numerical measurements

2 To measure the quantity ∆AH0, a numerical simulation at resolution Nx = 4096 is per- formed in which particles are emitted from the source situated at position xS =[⇡, ⇡], with a maximum lifetime for the particles t = L , where L =2⇡ is the size of the pmax 2urms square domain. Table 3.1 gives the parameters used for the simulation as well as global flow quantities. Because we simulate discrete particles, we will of course not find PH at an exact distance r from P0. Limiting ourselves to the closest particle from the theoretical PH will then yield an error on AH and on ∆AH0. The maximum precision we can obtain for AH is equal to the time step ∆t because one particle is released per time step. One way to get more precision 52 CHAPTER 3. TRACERS DISPERSION

102 102 2b : r, rE ≫ lf -1 A0 > τr : ∆AH0 ≠ f(rE ) 2a : r, rE ≫ lf E τ ∆ -1.5 1 A0 < rE : AH0 ∝ r 10 101 1b : rE ≪ lf ≪ r -2 2/3 r −A0 ∆AH0 ∝ lf e f 3/2 3/2 r ∝ A ≪ ∆ f 0 r A0 : AH0 ≠ f(r) -2.5

r/l 0 10 r/l 100 ↑ -3 r = lf ← τl f -3.5 ≪ -1 1a : r, rE lf -1 10 ∆ 10 AH0 ∝ r -4

-4.5 10-2 10-1 100 10-2 10-1 100 A0/TI A0/TI

Figure 3.11: Left: scheme of expected regimes for ∆A . Notation = f( ) means: ”is h H0i 6 · not a function of”. Right: contour plot of log ∆A . In the simulation, the condition 10 h H0i r l ∆A l /u =4 10−3. E � f , H0 � f rms ⇥

⌫ ∆t φS LTI urms " kf lf tlf 4 10−4 10−5 105 2⇡ 1.96 1.6 41.6 1000 2⇡ 10−3 0.01 ⇥ ⇥ Table 3.1: Some parameters of the simulation. ⌫ is the fluid molecular viscosity. ∆t is the time step. φS is the injection rate in units of particles injected per time step. L =2⇡ is the size of the square domain. TI = L/urms is the time associated to the integral scales. urms = p2E is the mean square velocity. " = 2⌫Z. kf is the forcing wave-number. −3 −1/3 2/3 − lf = 10 L is the forcing scale. tlf = " lf is the time associated to the forcing scale. could be to increase the number of emitted particles per time step and to interpolate linearly their age between the emission time t and t ∆t. This would however increase E E − the numerical cost, and another method is implemented. Let D(PA,PB) be the distance between particles PA and PB. Then particle PH cor- responds to a fictive point such that D(P0,PH )=r whose age is linearly interpolated between particles, say PH1 and PH2 , which are directly surrounding particle PH , i.e for which D(P0,PH1 ) r. This is tantamount to suppose that particles in the continuous framework are aligned between PH1 and PH2 , or equivalently that the line is not deformed on a time scale ∆t. This is a valid hypotheses if D(PH1 ,PH2 ) is small compared to the correlation length of the flow, and if ∆t is small compared to the temporal correlation of velocity increment associated with the scale D(PH1 ,PH2 ). The determination of AH then reads:

2 2 rH ·rH1 rH rH2 + rH rH2 2 k − k k − k − krH k rH1 A = A +(A A )r k k (3.51) H H1 H2 − H1 r r k H1 − H2 k 3.3. RESULTS 53

2r 2r

r PH2 r PH PH2 P H1 PH P = P P0 0 H1

Figure 3.12: Left: the position of the true particle PH has to be interpolated between two adjacent discrete particles. Right: For A T and / or r u ∆t, the low spatial 0 � I ⌧ rms resolution of particles along the line yields a very bad estimation of its length.

For A T and / or r u ∆t, particles separate quickly and the distance r and 0 � I ⌧ rms 2r are rapidly attained. In that case, particle P0 corresponds to PH1 and particles P0, PH and P are aligned (see Figure 3.12). In this situation, ∆A trivially scales as r. This H1 h H0i situation corresponds to a serious breaking of the continuous approximation at the scale r, and these events are removed from the statistics. As a consequence, values of ∆A will h H0i be lower-bounded by ∆t.

The simulation is run for 2 TI after the number of particles has reached its equilibrium. The quantity ∆AH0 is determined for all A0 and various r logarithmically spaced between −4 10 = 20⇡lf and 1 = L/2⇡. This is performed for 200 different instantaneous distributions of tracers over which it is averaged. We now wish to verify the theoretical predictions given above. Figure 3.13 (left panel) shows the age difference A /T as a function of the distance r/l for various A . Relation h H0 i I f 0 1a, ∆A r, is indeed recovered in a wide range of scales for A T and r l . h H0i/ 0 ⌧ I ⌧ f As A grows, the asymptotic regime for A r2/3, case 1b, is recovered, and ∆A is 0 0 � h H0i independent of r. The breaking of the lines correspond to the break-up of the continuity of the line at the scale r (see Figure 3.12 (right)). 3/2 Figure 3.13 (right panel) displays the same quantity as a function of (r/lf )/A0 . In the explosive Richardson separation, r(t) A3/2 is a quantity independent of the initial / 0 separation r , or equivalently that ∆A = f(A ). This is verified in our measures where E h H0i6 0 such a point is indeed observable in Figure 3.13 at ((r/l )/A3/2, ∆A ) (2 102, 2 f 0 h H0i ⇡ ⇥ ⇥ 10−3). A contour plot of the quantity log ∆A represents these various regimes along with h H0i the theoretical predictions in Figure 3.11 (right). 54 CHAPTER 3. TRACERS DISPERSION

100 100 A0/TI =0.0015 A0/TI =0.3867 10-1 10-1

A0 -2 10-2 10 ⟩ ⟩ 0 0 H H A A ∆ ∆ -3 -3 ⟨ ⟨ 10 10

-4 10-4 10 A0/TI =0.0015 A0/TI =0.2908

0 2 4 10-2 100 102 10 10 10 3/2 r/lf (r/lf )/A0 Figure 3.13: Age difference between particles separated from a distance r for various ages A0 of the reference particle. The range of A0 spans the whole values available until TI , monotonically increasing in the direction indicated by the arrow. The black dashed line has a slope 1 and represents ∆A r. H0 /

3.4 Brief conclusion

We have analysed the dispersion of tracer particles continuously emitted from a point source. This issue targets various natural phenomena such as the release of polluting species at ocean surfaces. The turbulent mixing following the release results from a com- plex combination. Numerical simulations of two-dimensional turbulence have been carried. One point quantities, such as mean square displacements and average radial concentra- tion profiles were shown to obey simple ballistic dynamic at short times from emission and diffusion after one Lagrangian velocity correlation time scale. Then, we observed that the combination of various mixing processes, involving particles of very different ages, doesn’t allow for the description of mass fluctuations with fractal dimensions. Another phenomenological approach was then proposed to account for the mixing of temporal and spatial correlations. It makes use of the knowledge of relative pair separation as a function of particles emission time and initial separation. Part II

Inertial particle-laden flows

55

CHAPTER 4

A lattice method for the numerical modelling of inertial particles

Contents 4.1 Inertial particles dynamics ...... 57 4.1.1 Individual particles ...... 57 4.2 The modelling of dispersed multiphase flows ...... 61 4.2.1 From microscopic description to macroscopic quantities ...... 62 4.3 Description of the method ...... 67 4.4 Application to a one-dimensional random flow ...... 70 4.4.1 Particle dynamics for d =1 ...... 70 4.4.2 Lattice-particle simulations ...... 74 4.5 Application to incompressible two-dimensional flows ...... 79 4.5.1 Cellular flow ...... 79 4.5.2 Heavy particles in 2D turbulence ...... 81 4.6 Conclusions ...... 86

4.1 Inertial particles dynamics

4.1.1 Individual particles Massive particles, i.e. whose density differs from that of the fluid, experience a large variety of forces exerted by the carrier flow. Considering large particles requires to integrate the total constrain exerted by the fluid from the non-linear Navier-Stokes equations on the

57 58 CHAPTER 4. LATTICE PARTICLES METHOD surface of the particle. This requires integrating analytically the full velocity field, which is in general not possible. However, an explicit expression for the total force can be derived in the case of small particles, i.e. smaller than the smallest characteristic length scale of the flow, and in the case of small velocity difference with surrounding flow. The flow around the particle is then in a laminar state, and the Reynolds number associated to the particle, Re = d V u /⌫, with d is the particle diameter and ⌫ the fluid viscosity, is low. p p h p − i p The non-linear term in the Navier-Stokes equation may then be neglected and the Stokes equation integrated around the particle (see Maxey & Riley (1983) for details). This leads to a closed equation that we rewrite here in order to illustrate the complexity of the forces acting on each individual particle in a velocity field u(x): dV Du 1 dV Du m p =(m m )g + m (X ) m p (X ) p dt p − f f Dt p − 2 f dt − Dt p ✓ ◆ t 2 d⌧ d 6⇡dp⇢f ⌫ (vp u(Xp(⌧), ⌧)) − ⇡⌫(t ⌧) d⌧ − Z0 − 6⇡d ⇢ ⌫ (V p u(X )) . (4.1) − p f p − p mp denotes the mass of each individual particle, Vp its velocity and Xp its position. ⌫ is the kinematic viscosity of the carrier flow and mf displaced mass of fluid by the particle. The first term on the right-hand side is the buoyancy force. The second is the ac- celeration of the unperturbed flow at the particle position. The third one is the inertial correction, which arises because of the displaced flow by the particle, accounting for an ad- ditional transported mass. The fourth term is the Basset-Boussinesq history force, which obviously add a complicated effect linked to the past of the trajectory. It is due to the particle wake which acts to diffuse the flow vorticity apart from the particle trajectory. It is generally neglected when considering very small particles, because in that case the wake is dissipated on a sufficiently short length scale. The last term is the Stokes viscous drag. In this study, gravity is neglected, which is the case when fluid accelerations are stronger than g and particles not too massive. With the simplifications cited above, one gets:

dVp Du 1 = β (Xp(t),t) [Vp u(Xp(t),t)] . (4.2) dt Dt − ⌧p −

The first term on the right is the added-mass factor, with β =3⇢f /(⇢f +2⇢p). For heavy particles (⇢ ⇢ ), we get β 1 and this term is also neglected. The parameter p � f ⌧ ⌧ = d2/(3β⌫) denotes the response time of the particles. For ⇢ ⇢ , ⌧ =2⇢ d2/(9⇢ ⌫). p p p � f p p p f Finally, the dynamic of one particle, Xp(t), obeys the following equations:

X˙ p(t)=Vp(t), (4.3) 1 V˙p(t)= [Vp(Xp,t) u(Xp,t)] . (4.4) −⌧p − 4.1. INERTIAL PARTICLES DYNAMICS 59

The dots denote temporal derivatives. The non-dimensional number quantifying the relative inertia, the Stokes number, is built by the ratio of this response time and a relevant characteristic timescale of the flow: ⌧ St = p (4.5) ⌧f

Fundamental differences characterise the asymptotics of very low and large inertia. One noticeable fact about inertial particles in fluids, at least for moderate inertia, is that they exhibit clustering, i.e. they concentrate in regions of given carrier flow topology. The knowledge, quantification and prediction of such clustering is of central importance when considering situations where interactions between particles play a key role, such as in coalescence, or advection-reaction (Bodenschatz et al., 2010; Krstulovic et al., 2013). The dynamic described by (4.3) and 4.4 is dissipative, with an associated phase-space contraction rate of d/⌧ . It has been shown in Bec (2003) that the long time behaviour − p of those particles converge toward a multifractal set.

St 1 ⌧ In the limit when St 0, particles behave like tracers, and the difference V u in (4.4) ! − cancels. One may then track individual particles in a Lagrangian framework, or consider solely a transport equation for their density ⇢p(x) by the velocity vp(x)=u(x). The fractal dimension of the ensemble is the space dimension = d: the ensemble is homogeneous in D space. For very small response time (i.e. fast relaxation), the same technique may be used, but with a small correction applied to vp of the order of ⌧p (Maxey, 1987):

Du v = u ⌧ . (4.6) p − p Dt which appeared to yield good results compared to Lagrangian simulation for up to St 0.2 ⇡ (see, for instance, Shotorban & Balachandar (2006)). Increasing St, trajectories of particles and fluid elements begin to separate and an evo- lution equation for the particle density must be provided. This density ⇢p(x) is transported by a compressible flow (Balkovsky et al., 2001) and its evolution, along with the field vp(x), obeys the following dynamical system:

@ ⇢ + (v ⇢ )=0, (4.7) t p r · p p v = u + ⌧ (@ u +(u r)u) . (4.8) p p t · This formulation is valid for Stokes up to St (1) (Balachandar & Eaton, 2010), and ⇠ O is to be preferred when particles are not in close equilibrium with the carrier phase, for 60 CHAPTER 4. LATTICE PARTICLES METHOD

example when they are injected perpendicularly to the flow. The divergence of the velocity field is given by (Maxey, 1987):

2 2 1 @ui @uj @ui @uj r vp = − 8 + 9 . (4.9) · 4A > @xj @xi − @xj − @xi > <>✓ ◆ ✓ ◆ => strain rate vorticity > > > > with A a dimensionless parameter:| scaling{z as St−}1.| The relation{z (}4.9; ) indicates that the particles converge (the divergence is negative) where the strain dominates the vorticity, resulting in a preferential sampling of the high-strain regions. This effect increases with the Stokes number. Indeed, when looking at an instantaneous spatial repartition of the particles, those leave near-empty regions whose size increases with St (Goto & Vassilicos, 2006). However, this increase in size is not only due to the centrifugal effect. In two dimensions, it has been shown that, for all St < T/⌧⌘, where T is the characteristic timescale associated with the sweeping by the large eddies and ⌧eta the one associated to the dissipation scale, small inertial particles move away from non-zero acceleration zones. The clustering is better described by the correlation between the high- valued particle number density regions and the zero acceleration stagnation points (Chen et al., 2006a). Equations (4.8) and (4.7) have been successfully applied to study the dynamical prop- erties of weakly inertial particles. For example in Boffetta et al. (2007), they are used to determine fractal dimension of the phase space attractor and yielded good agreement with Lagrangian simulations.

St 1 � When the relaxation time gets larger, the existence and the uniqueness of the velocity field vp(x) is not guaranteed any more. Indeed, inertia causes particles to detach from fluid trajectories with the possibility to cross each other with different velocities at the same position x, forming caustics. Actually, this phenomenon already occurs for St (1). ⇠ O Figure 4.1 illustrates this phenomenon schematically in one dimension for a continuous line of particle. Obviously, this situation is predominant in the case where the fluid exhibits large and persistent gradients u. The rate of formation of those caustics depends also r on St. For example, in the regime of St 1, it has actually been shown that the rate ⌧ at which they form is exp( 1/(6St)) for ⌧ T ⌧ exp(1/6St)) (Derevyanko et al., / − p ⌧ ⌧ p 2007), thus showing an rapid decrease in the limit of vanishing inertia. This particularity of inertial particles forces one the describe the particle dynamics in a higher-dimensional space. In the limit St , the motion is totally ballistic, with V independent of u. The !1 p fractal dimension is =2d, and particles occupy the full position-velocity phase-space. D 4.2. THE MODELLING OF DISPERSED MULTIPHASE FLOWS 61

v !(#, %)

'(#,()

x

Figure 4.1: Illustration in one dimension of caustic formation: when inertia is large, the particle density f may overshoot the fluid velocity profile u and form multivalued velocity regions (blue-shaded on the figure).

In between the asymptotics St 0 and St , a maximum of clustering can be found, ! !1 characterised by a minimum fractal dimension. This is depicted by Figure 4.2 (right), where the dimension between the Lyapunov and the space dimension is represented as a function of St for d = 2 and d = 3. Figure 4.2 (left) illustrates the particles distribution near the maximum of clustering (St 0.2). ⇠ An analysis of the domain of validity for each approximation in terms of St may be found in (Balachandar & Eaton, 2010).

4.2 The modelling of dispersed multiphase flows

Many processing technologies require good analysis of their capabilities and performance. Examples include cavitating pumps, papermaking, fluidized beds, etc. Very few processes involving material transport don’t benefit from a better understanding of a multiphase dy- namics. The definition of multiphase flows encompasses diverse cases, such as an arbitrary number of fluid mixtures, or solid suspensions. In addition, suspensions may involve parti- cles of the same kind but of unique size (we then talk about monodisperse suspensions) or of various sizes (polydisperse ). To add up in difficulty, coupling between different phases may also be considered. This increases the degree of complexity depending on the number of phases and the nature of their interactions. Furthermore, the presence of a dispersed phase can make appear different scale-resolving issues that would not be present when considering the fluid phase alone. Think for example of bubbly flows, where bubbles may leave along their trajectory a turbulent wake, while the large-scale flow is actually laminar (Mudde et al., 2008). 62 CHAPTER 4. LATTICE PARTICLES METHOD

0.5 100

−2 0.25 10 d = 2 d = 2 slope = 2 − d = 3 10 4 −3 −2 −1 d 10 10 10 −

0 L d d = 3

−0.25

−0.5 0 0.1 0.2 S 0.3 0.4 0.5 η

Figure 4.2: Left: Illustration of inertial particles distribution in position space. St = 10−2. From Bec (2005). Right: difference between fractal dimension of the particle set and the space dimension d for increasing St (from Bec (2003)).

All of those difficulties add up, taking into account that most natural and industrial flows are in a turbulent state. And already, direct numerical simulations of a single tur- bulent phase need to resolve a tremendously large scale separation to reach high Reynolds numbers (up to 106 in some fluidized-beds). For all those reasons, the direct numerical simulation (DNS) of realistic multiphase flows is non attainable with any nowadays com- putational performance. Nevertheless, they constitute a required tool to better understand physical mechanisms and thus improving and validating the approximations made. As an example, models for polydisperse suspensions benefit from the knowledge of the interaction between particles of different sizes (Tenneti et al., 2010; Yin & Sundaresan, 2009). Such difficulties lead scientists and engineers to develop more and more sophisticated models and this section is dedicated to introduce briefly some of them.

4.2.1 From microscopic description to macroscopic quantities

Basically, one can think about two classes of methods to model multiphase flows. One is fundamentally relying on the detailed physical interactions between solid and fluid particles, or between particles and solid interfaces. The other is more phenomenological, based on conservation laws and constitutes a hydrodynamic approximation in which all relevant quantities are treated as continuous space fields. The term Lagrangian is used for models considering the evolution and tracking of discrete elements, while the term Eulerian is used 4.2. THE MODELLING OF DISPERSED MULTIPHASE FLOWS 63 for continuous fields approaches. The transition between these two descriptions can be achieved for example through ensemble averaging: microscopic dynamical equations are volume or ensemble averaged to yield dynamical equations for the low order macroscopic quantities, such as mass or momentum (Drew & Passman, 1999). In between microscopic and macroscopic models come the kinetic equations whose so- lution yields a function f defined over a phase space of relevant mesoscopic variables (vol- ume, velocity, elasticity, etc.). Approximations are made from the microscopic description to model physical interactions. Detailed physical studies and direct numerical simulations are used to determine how the mesoscale variable associated to a given particle will be affected by the external forces, fluid, other particles (collisions), etc. The moments of the kinetic equation allow then for recovering the macroscopic con- servation equations. A famous example of kinetic equation is the one for gas dynamics or Boltzmann equation (Cercignani, 1988). In this chapter, the collisions between solid particles are neglected and the mesoscopic variables are limited to position and velocity.

Lagrangian simulations

A solution to integrate the particle dynamics is to resolve explicitly equations (4.3) and (4.4) by considering point particles. This method presents two big advantages: it is not restricted in terms of St, and because each point is treated individually, polydispersity may be included easily. However, for a too-large number of particles in high-Reynolds-number turbulent flows, the number of degree of freedom is huge and models must be introduced. Another limitation of this approach is the difficulty to design numerical methods for the back-reaction from the solid phase on the continuous phase. Indeed, one needs a large number of particles and several approximations to handle correctly the distribution of the particles reaction on the numerical grid-points of the continuous phase. This issue is discussed in chapter 5. It was actually one of the main motivation for the development of the Eulerian description of inertial particles presented in this chapter, as it yields a more natural treatment of particle retro-action on the fluid. When the number of particles becomes too large, so that their evolution is not com- putationally tractable, other stochastic approaches have to be considered. One of them is to build parcels which can be thought of as encompassing a large number of the discrete particles. These parcels have properties driven by a set of stochastic differential equations and their evolution represent Monte-Carlo simulations of the underlying pdf. Their properties form a state vector L obeying a Langevin equation. For example, in the simplest case for which L =(Xp, Vp, us(Xp)), where us is a velocity seen by the parcel, 64 CHAPTER 4. LATTICE PARTICLES METHOD the equation takes the following form (Minier & Peirano, 2001):

dXp = Vpdt, (4.10)

dVp = g(t, L)dt, (4.11)

dui = hi(t, L)dt + Bij(t, L)dWj, (4.12) Xj

where g is in our case the Stokes drag, h models the interaction between us and the mean fluid velocity u . B is the Wiener process amplitude encompassing the model for the h i fluctuating part of the turbulent fluid velocity, like a k ✏ model. The importance of the − consistency between the turbulence models for the fluid and solid phases has been stressed in Chibbaro & Minier (2011). These methods are largely applied in the engineering community. For example, Chib- baro & Minier (2008) have used this approach coupled to RANS equations for the fluid phase to study particle wall deposition and obtained a good agreement with experimental results.

Phase space description

The kinetic formulation for a particle population allows to encompass the physical proper- ties of the particles (position, velocity...) in terms of an Eulerian field in the phase space:

f(x, v,t)= δ(X (t) x)δ(V (t) v), (4.13) p − p − p X where δ is the delta function. With such a definition, the units of f are L−2dT d. The evolution equation for f may be obtained the following way. f being distributional, we make use of a test function. Let ' C1 be such a test function, infinitely smooth and 2 4.2. THE MODELLING OF DISPERSED MULTIPHASE FLOWS 65 with compact support. Then:

@t f(x, v,t)'(x, v)dx dv Z

= @t '(Xp(t), Vp(t)) " p # X = X˙ (t) r ' + V˙ (t) r ' p · x p · v p X h i = [V (t) r ' + A (t) r '] p · x p · v p X = v r '(x, v)δ(x X )δ(v V )dx dv · x − p − p p X Z + a r '(x, v)δ(x X )δ(v V )dx dv · v − p − p p X Z = r ' (vf)+r ' (af)dx dv x · v · Z [@ f + r (vf)+r (af)] ' dx dv =0 ) t x · v · Z where the integrals are taken over the phase-space and Ap is the particle acceleration. Integration by parts has been used for the last line. Because the last integral doesn’t depend on ', it comes:

@ f + r (vf)+r (af)=0. (4.14) t x · v · a = F /mp is introduced to denote the instantaneous acceleration of the particle. Equa- tion (4.14) is a Liouville equation expressing a conservation of the phase space density, whose dimension is L−2dT d. The second term on the right hand side is the streaming of the particles by the flow while the third term is the change in velocity due to the application of the force F . The knowledge of f(x, v) allows one to evaluate quantities such as density and mo- mentum in the position space. These are obtained by evaluating the first moments of the distribution f:

th d 0 : number density np(x)= f(x, v)d v (4.15) ZV st d 1 : momentum np(x)vpi (x)= vi f(x, v)d v (4.16) ZV nd d 2 : kinetic energy np(x)eij(x)= vivj f(x, v)d v (4.17) ZV 66 CHAPTER 4. LATTICE PARTICLES METHOD with the velocity domain. n (x) is the density in terms of number of particles. For V p a monodisperse non reacting phase as will be our case, we can define a mass density by ⇢(x)=mpnp(x) with a constant mp. A model is needed for the different terms appearing in the kinetic equation (4.14). In our case, as we make use of a phase-space only constituted by positions and velocities, only a model for the force has to be provided. In this chapter, we only consider dilute suspensions so that inter-particle collisions are neglected. For such suspensions, F reduces to a Stokes drag F = ⌧ −1(u v ) (see Section 4.1.1). p − p Solving the kinetic equation Starting from the kinetic equation (4.14), one has several strategies to solve for the distri- bution f. One is to integrate it over velocities, once it has been multiplied by vn which yields evolution equations for the n first moments of velocity. Evolution equations for the macroscopic quantities are then recovered. However, it can be shown that the equation for a moment of order n involves the (n + 1)-th order. This forms a set of infinite equations yielding an unclosed hierarchy, and this is why closure is necessary at some order n. For example, if M i denotes the moment of order i, being itself a tensor of order i, the following evolution equations are (Fox, 2012):

@M 0 @M 1 + i =0, @t @xi 1 2 @Mi @Mij 1 0 1 + = (M ui Mi ), @t @xj ⌧p − 2 3 @Mij @Mijk 1 1 1 2 + = (Mi uj + uiMj 2Mij). (4.18) @t @xk ⌧p −

This system is said to be unclosed because one needs an evolution equation for M 3, which would make appear M 4, etc. A closed formulation for M 3 is thus needed, which may be built by combining the lower order moments. The quality of the closure may itself be optimised and improved by relevant knowledge of the underlying detailed physics. As already stated, one must take care that the dynamics of large Stokes numbers parti- cles require a description in the 2d phase-space (x, v) in order to resolve velocity dispersion. To account for this dispersion, some methods involve for example the integration of the moments up to the second order, then a reconstruction of the full distribution f(x, v). This distribution is then used to compute the third order moment M3 to be used in (4.18). This method however requires to assume the functional form of f (see, for instance, Simonin et al. (1993)). In Aguinaga et al. (2009), a closed kinetic equation is developed where the interaction between particles and turbulence is modelled through a return-to-equilibrium term similar to the Bhatnagar–Gross–Krook model (Bhatnagar et al., 1954), namely Ω = ⌧ −1(f f ), i − − p 4.3. DESCRIPTION OF THE METHOD 67 where fp is an equilibrium distribution and ⌧ the relaxation time. Different models exist then for fp, including a Gaussian pdf. The resulting equation may then be explicitly solved using finite differences (Aguinaga et al., 2009) or via a Lattice Boltzmann scheme (Fede et al., 2015). The following sections describe the part of this thesis dedicated to this issue. A novel approach to solve the kinetic equation is presented, based on the integration of the distri- bution f in the full position-velocity phase space. Some of the results presented here may be found in the paper published in Comptes Rendus de M´ecanique (Laenen et al., 2016). In this paper, the considered kinetic equation differs from (4.14) by an additional diffusive term, i.e it reads: @ f + r (vf)+r (af)  2f =0. (4.19) t x · v · − vrv

4.3 Description of the method

The solutions f(x, v,t) to the Liouville equation (4.14) are defined in the full position- velocity phase-space Ω Rd, where Ω designates a d-dimensional bounded spatial domain. ⇥ To simulate numerically the dynamics, we divide the phase-space in (2 d)-dimensional ⇥ hypercubes. We then approximate f(x, v,t) as a piecewise-constant scalar field on this lattice. Positions are discretised on a uniform grid with spacing ∆x in all directions. In principle, f has to be defined for arbitrary large velocities. We however assume that relevant values of v are restricted to a bounded interval [ V ,V ]d , where V has to be − max max max specified from physical arguments based on the forces applied on the particles. Velocities d F ∆ are assumed to take Nv values, so that the grid spacing reads v =2Vmax/Nv. Figure 4.3 illustrates the phase-space discretisation in the one-dimensional case with Nv = 5. The various cells in position-velocity contain a given mass of particles. All these particles are assumed to have a position and velocity equal to that at the centre of the cell. The three phase-space differential operators appearing in equation (4.14), namely the advection, the particle forcing, and the diffusion, are applied one after the other, following an operator splitting method (LeVeque, 2002). For the advection step, we use a technique inspired from the Lattice-Boltzmann method (see Succi (2001)). The time stepping is cho- sen so that a discrete velocity exactly matches a shift in positions by an integer number of grid-points. Namely, we prescribe ∆x = ∆v ∆t. All the particle phase-space mass located in [ ∆v/2, ∆v/2] does not move; that in [∆v/2, 3∆v/2] is shifted by one spatial gridpoint − to the right and that in [ 3∆v/2, ∆v/2] to the left, etc. All the mass is displaced from one − − cell to another according to its own discrete velocity value. This evolution is sketched by black horizontal arrows in Figure4.3. This specific choice for the time-stepping implies that the advection (in space) is treated exactly for the discrete system. The next steps consist in applying the force acting on the particles and the diffusion. The corresponding terms in equation (4.14) are conservation laws, which suggests using a finite-volume approxima- tion. The time evolutions due to forcing and diffusion are performed successively. In both 68 CHAPTER 4. LATTICE PARTICLES METHOD

+V max

v u(x,t)

∆ v −V ∆ x x max

Figure 4.3: Sketch of the lattice dynamics in the (x, v) position-velocity phase space. The solid curve is the fluid velocity profile; the grey-scale tiling represents the discretisation of particles mass in phase space. The black horizontal arrows show advection, while the blue and red vertical arrows are forcing and diffusion, respectively. cases, we use classical schemes (see below), where zero-flux conditions are imposed on the boundary of [ V ,V ]d. The force is evaluated using the values of v at the centres of − max max the cells and f is approximated using finite differences. These steps are illustrated by rv the horizontal blue and red arrows in Figure 4.3. A few comments on the convergence and stability of the proposed method. Clearly, except for specific singular forcings, all the linear differential operators involved in (4.14) are expected to be bounded.1 We can thus invoke the equivalence (or Lax–Richtmyer) theorem for linear differential equations that ensures convergence, provided the scheme is stable and consistent LeVeque (2002). For the operator associated to particle acceleration, we use in this study either a first- order upwind finite-volume scheme or a higher-order flux limiter by following the strategy proposed in Hundsdorfer et al. (1995). The upwind scheme is first-order accurate and is well-known for being consistent and stable if it satisfies the Courant–Friedrichs–Lewy (CFL) condition. This requires that the time needed to accelerate particles by the grid size ∆v is larger than the time step ∆t, leading to the stability condition

CFL = (∆t/∆v) max (x, v,t) /mp < 1. (4.20) x,v,t |F | The upwind scheme is however known to suffer from numerical diffusion, and obviously, one should only expect to recover the correct dynamics only when the numerical diffusion

1Notice that, although the velocity might explicitly appear in the force F, we only solve for a compact domain of velocities, thus preventing divergences. 4.3. DESCRIPTION OF THE METHOD 69

num is much smaller than the physical one . The average numerical diffusion can be estimated as  ∆t/∆v. To limit the effects of this numerical diffusion, we h v,numi⇡hFi have also used a flux-limiter scheme. While taking benefit of a higher-order approximation where the field is smooth, it uses the ratio between consecutive flux gradients to reduce the order in the presence of strong gradients only. The limiter is a nonlinear function of the phase-space density field and the stability is ensured provided that it is total-variation diminishing (TVD), see LeVeque (2002). Among the various available TVD limiters, we choose the scheme proposed in Koren (1993) with parameter 2/3. For the term associated to diffusion, the flux at the interface between two velocity cells is computed using finite differences. The resulting finite-volume scheme is thus equivalent to compute a discrete Laplacian on the velocity mesh. The stability condition is then given by  ∆t 1 v < . (4.21) ∆v2 2 To summarize, the stability and convergence of the proposed method is ensured when both (4.20)and(4.21) are satisfied. From now on we restrict ourselves to small and heavy particles whose interaction with the carrier fluid is dominated by viscous drag and diffusion. In that case, we have that the acceleration ap of one particle reads:

dvp 1 ap = = (vp u(xp,t)) + p2v η(t), (4.22) dt −⌧p − where η(t) is the standard d-dimensional white noise and the fluid velocity field u(x,t) is prescribed and assumed to be in a (statistically) stationary state. This Stokes drag involves 2 the viscous particle response time ⌧p =2⇢pa /(9⇢f ⌫), with a the radius of the particles, ⌫ the viscosity of the fluid, ⇢ ⇢ the particle and fluid mass densities, respectively. Inertia p � f is quantified by the Stokes number St = ⌧p/⌧f , where ⌧f is a characteristic time of the carrier flow. The diffusion results from the random collisions between the considered macroscopic particle and the molecules of the underlying gas. Assuming thermodynamic equilibrium, the diffusion coefficient reads  =2kB T/(mp ⌧p), where kB is the Boltzmann constant and T the absolute temperature. The effect of diffusion is measured by the non-dimensional number = ⌧ /U 2 (U being a characteristic velocity of the fluid flow). K f f f Such a specific dynamics leads to appropriate estimates for the bound Vmax in par- ticle velocity. One can indeed easily check that when  = 0, we always have v | p|  maxx u(x,t) . In a deterministic fluid flow, as for instance when u is stationary, this ,t | | gives the natural choice V = maxx u(x,t) . However, in most situations, the maximal max ,t | | fluid velocity is not known a priori. One then relies on the statistical properties of u, as for instance its root-mean square value u = u2 1/2. Usually the one-time, one-point rms h i i statistics of fluctuating velocity fields (being random or turbulent) are well described by a Gaussian distribution. This ensures that by choosing Vmax =3urms, the probability that 70 CHAPTER 4. LATTICE PARTICLES METHOD a particle has a velocity out of the prescribed bounds is less than 1%. Such estimates are rather rough. In practice, it is known that the typical particle velocity decreases as a function of the Stokes number. It was for instance shown in Abrahamson (1975) that v 2 u2 /St at very large Stokes numbers. An efficient choice for V should account h| p| i/ rms max for that. The equation solved in the velocity space is analogous to an advection equation in position space. In finite volume and finite difference methods, those are known to suffer from numerical diffusion and to introduce a non physical broadening of the solution along the diffusive dimension, i.e. an increase of the variance of the distribution. For example, the simple upwind scheme yields a numerical diffusion term with diffusivity equals to  (v)=(1 a(v)∆t ) a(v)∆v . For more details, see, for instance, Cushman-Roisin & num − ∆v 2 Beckers (2011).

(a) Nx = 16384,Nv = 127 (b) Nx = 4096,Nv = 33 (c) Nx = 1024,Nv =9

Figure 4.4: Illustration of the effect of numerical diffusion. Phase space resolutions Nx and N are increased at constant ∆t with no physical diffusion ( = 0) . v K The importance of this numerical diffusion can be quantified by the grid Peclet number ∆va Pe = rms . (4.23) num  It is thus of importance that  . num ⌧ In the next two sections we investigate two different cases: first a one-dimensional random Gaussian carrier flow with a prescribed correlation time and, second, a two- dimensional turbulent carrier flow that is a solution to the forced incompressible Navier- Stokes equations.

4.4 Application to a one-dimensional random flow

4.4.1 Particle dynamics for d =1 In this section, our method is tested in a one-dimensional situation. For that, we assume that the fluid velocity is a Gaussian random field, which consists in the superposition of 4.4. APPLICATION TO A ONE-DIMENSIONAL RANDOM FLOW 71 two modes whose amplitudes are Ornstein–Uhlenbeck processes, namely

u(x, t)=A1(t) cos(2⇡ x/L)+A2(t) sin(2⇡ x/L) (4.24) dA (t) 1 2 i = A (t)+ ⇠ (t) (4.25) dt −⌧ i ⌧ i f r f where the ⇠ ’s are independent white noises with correlations ⇠ (t) ⇠ (t0) = u2 δ(t t0). i h i i i rms − This flow is by definition fully compressible (potential) and spatially periodic with period L. It is characterized by its amplitude (u(x, t))2 1/2 = u and its correlation time ⌧ , h i rms f which are fixed parameters. We focus on the case when the Kubo number Ku = ⌧f urms/L is of the order of unity. We next consider particles suspended in this flow and following the dynamics (4.22). The relevant Stokes number is then defined as St = ⌧p urms/L and the relative impact of diffusion is measured by = L/u3 . When diffusion is neglected ( 0), the particles K rms K! distribute on a dynamical attractor (see Figure 4.5 Left) whose properties depend strongly on St. These strange attractors are typically fractal objects in the phase space and they

(a) =0 (b) =2⇡ 10−4 (c) =2⇡ 10−3 K K ⇥ K ⇥ Figure 4.5: Instantaneous snapshots of the particle positions in the (x, v) plane for St 2 ⇡ for varying diffusivities . The folded structures are spread out by diffusion. K are characterized by their fractal dimension spectrum (Hentschel & Procaccia, 1983). The locations of particles are obtained by projecting theses sets on the position space and might thus inherit the associated clustering (Bec, 2003). The dimension that is relevant for binary interactions between particles is the correlation dimension , which relates to D2 the probability of having two particles within a given distance, namely

p<(r)=P( X(1)(t) X(2)(t)

a function of the Stokes number are displayed in the inset of Figure 4.6. indeed varies D2 from 0 at small Stokes numbers to values close to one. For St = 0, the particles concentrate on a point; their distribution is said to be atomic and = 0. This is a consequence of the D2 compressibility of the one-dimensional (potential) flow. Actually this behaviour persists for finite Stokes numbers, up to a critical value St?, as shown in Wilkinson & Mehlig (2003) in the case where ⌧ L/u (that is Ku 0). We observe here St? 0.6. For St > St?, f ⌧ rms ! ⇡ the dimension increases and tends to a homogeneous distribution ( = 1) at large particle D2 inertia. When one has only access to the Eulerian density of particles, the distribution of dis- tances cannot be directly inferred from (4.26). One then relies on the coarse-grained density of particles r/2 0 0 ⇢r(x, t)= dx dvf(x + x , v,t). (4.27) Z−r/2 Z It is known that, under some assumptions on the ergodicity of the particle dynamics, the second-order moment of this quantity scales as ⇢2 rD2−1 (see, e.g., Hentschel & h ri/ Procaccia (1983)). In one dimension, this second-order moment is exactly the same as the radial distribution function. This quantity will be used in the next sections to address the physical relevance of the lattice-particle method. It is of particular interest when considering collisions between particles. Indeed, as explained for instance in Sundaram & Collins (1997), the ghost-collision approximation leads to write the collision rate between particles as the product of two contributions: one coming from clustering and entailed in the radial distribution function, and another related to the typical velocity differences between particles at a given distance. This second quantity relates to the particle velocity (first-order) structure function

S (r)= V (1) V (2) X(1) X(2) = r . (4.28) 1 | p − p | | p − p | D � E This is the average of the amplitude of the velocity� difference between two particles that � are at a given distance r. As the probability of distances, this quantity behaves as a power law S (r) r⇣1 for r L (see e.g. Bec et al. (2005)). The exponent ⇣ , shown in the 1 ⇠ ⌧ 1 inset of Figure 4.6 decreases from 1 at St = 0, corresponding to a differentiable particle velocity field, to 0 when St , which indicates that particle velocity differences become !1 uncorrelated with their distances. Again, when working with the phase-space density one cannot use (4.28) but relies on dv dv0 f(x, v) f(x + r, v0) v v0 S (r)= | − | . (4.29) 1 0 0 ⌦R Rdv dv f(x, v) f(x + r, v ) ↵ As the second-order moment of the⌦R coarse-grainedR density, this↵ quantity will also be used as a physical observable for benchmarking the method. In the above discussion, we have neglected the effects of diffusion. It is for instance expected to alter clustering properties by blurring the particle distribution at small scales. 4.4. APPLICATION TO A ONE-DIMENSIONAL RANDOM FLOW 73

This is illustrated in Figure 4.5 where one can compare the instantaneous phase-space particle positions in the absence of diffusion (Left) and when it is present (Middle and Right) at the same time and for the same realization of the fluid velocity. At large scales, identical patterns are present, but diffusion acts at small scale and smoothes out the fine fractal structure of the distribution. One can easily estimate the scales at which this crossover occurs. Diffusion is responsible for a dispersion vd in velocities that can be ff 2 obtained by balancing Stokes drag and di usion in the particle dynamics, namely vd/⌧p , 1/2 1/2 ⇡ so that vd ⌧p  . This dispersion in velocity is responsible for a dispersion in positions ⇠ 3/2 1/2 3/2 1/2 on scales of the order of ` = ⌧ v ⌧p  = St L. Hence, when diffusion is small d p d ⇠ K enough and ` L, the spatial distribution of particles is unchanged by diffusion at length d ⌧ scales r ` , and the probability that two particles are at a distance less than r behaves � d as p<(r) (r/L)D2 . For r ` , diffusion becomes dominant, the particles distribute in a 2 ⇠ ⌧ d homogeneous manner and p<(r) rd, with d = 1 being the space dimension. By continuity 2 / at r = ` ,wegetp<(r) (` /L)D2 (r/` ) at small scales. d 2 ⇠ d d

100 K =0 −12 K =2π 10 D 10-1 − 2 K =2π 10 8 r K =2π 10−4 10-2

) d r -3 r 1.5 ( 10 2

p ζ1 1 D2 10-4 0.5

10-5 0 0123St 10-6 10-4 10-2 100 r/L < ff Figure 4.6: Cumulative probability p2 (r) of inter-particle distances for various di usivities  and for St 2. One observes at low diffusivities and for r>` a behaviour (r/L)D2 ⇡ d / with 2 0.7

This picture is confirmed numerically as shown in Figure 4.6 which represents the scale- < ff behaviour of p2 (r) for a fixed Stokes number and various values of the di usivity . One clearly observes the homogeneous distribution rd at small scales and the fractal scaling / rD2 in an intermediate range. The predicted transition between the two behaviours / 74 CHAPTER 4. LATTICE PARTICLES METHOD is indicated by the vertical lines at the diffusive scale `d. A homogeneous distribution is recovered for r . `d/10. Velocity statistics are also altered by the presence of diffusion. The structure function S (r) is expected to behave as r⇣1 for ` r L and to saturate to a constant value 1 d ⌧ ⌧ when r ` . By continuity, the value of this plateau should be `⇣1 ⇣1/2. Note ⌧ d ⇠ d ⇠ K finally that the slow convergence ` /L p as 0 implies that very small values of d / K K! the diffusion are needed in order to clearly recover the statistics of diffusive-less particles as an intermediate asymptotics.

4.4.2 Lattice-particle simulations We now turn to the application of the lattice-particle method described in Section 4.3 to this one-dimensional situation. We compare the results to Lagrangian simulations where we track the time evolution of Np particles randomly seeded in space with zero initial velocity. We choose and normalize the initial phase-space density f(x, v, 0) to match the Lagrangian settings. The distribution is uniform over the cells, concentrated on a vanishing ∆ ∆ velocity and the total mass is such that i,j f(xi,vj,t) x v = Np. In all simulations, the maximum velocity is set to V = u = 1 and we have chosen L =2⇡ and ⌧ = 1. max Prms f In these units, the time step is kept fixed at ∆t = ∆x/∆v =2−6⇡ 0.05. The number of n ⇡ discrete velocities is of the form 2 + 1 and is varied between Nv = 3 to 129. The number n+6 of spatial collocation points is then given by 2 and thus varies between Nx = 128 to 8192. Note that, because of the CFL condition (4.20), this choice restricts the number of discrete velocities that can be used to Nv < 1 + 128 St/(2⇡).

1 1

0.8

0.6

v 0

0.4

0.2

-1 0 0 π 2π x

Figure 4.7: Position-velocity phase-space positions of Lagrangian particles (black dots) on the top of the field obtained by the lattice-particle method (coloured background). The diffusivity is here  = 10−3 ( 6.28.10−3) and St 2. K ⇡ ⇡ Figure 4.7 represents simultaneously the phase-space distribution of Lagrangian parti- 4.4. APPLICATION TO A ONE-DIMENSIONAL RANDOM FLOW 75 cles and the numerical approximation obtained by the particle-lattice method for Nv = 129. Clearly, one observes that the method fairly reproduces the distribution of particles, in- cluding the depleted zones, as well as the more concentrated regions. Furthermore, the method is able to catch multivalued particle velocities. We have for instance up to three branches in v for x 3⇡/2. It is important to emphasize that numerical diffusion is of ' course present, and that it has to be smaller than the physical diffusion  in order for the approach to be consistent with the Lagrangian dynamics. To get a more quantitative insight on the convergence of the method, we next com- pare the coarse-grained densities obtained from the Lagrangian simulation and the lattice- L particle approximation of the phase-space density. The first, denoted ⇢r is computed by counting the number of particles contained in the different boxes of a tiling of size r. The E second is written as ⇢r and is obtained by summing over velocities and coarse-graining over a scale r the phase-space density obtained numerically. To confirm the convergence of the method, we measure for a fixed r the behaviour of the 2-norm of the difference between L E L ⇢r and ⇢r , namely 2 1/2 ⇢L ⇢E = ⇢L(x, t) ⇢E(x, t) , (4.30) k r − r k r − r where the angular brackets encompassD� a spatial and a� timeE average. Figure 4.8 shows h·i

100 2 St =1.3 St =1.6 1.8

∥ ∆ St =1.9 L r ∝ v 1.6 ρ St =2.2 ∥ /

-1 St =2.5 ⟩ N ∥ v =3 10 2 r E r ρ Nv

ρ 1.4 =5 ⟨ Nv =9 − N L r v = 17 ρ N ∥ 1.2 v = 33 Nv = 65 Nv = 129 -2 10 Lagrangian 1 101 102 10-4 10-3 10-2 10-1 100 Nv r/L

Figure 4.8: Left: Relative -error of the lattice-particle method for evaluating the coarse- L2 grained density ⇢r over a scale r = L/128 as a function of the number of velocity grid-points Nv and for various values of the Stokes number, as labelled. Right: Convergence of the second-order moment of the coarse-grained density ⇢2 , which is shown as a function of r h ri for St 1.9, = ⇡ 10−2, and various lattice velocity resolutions N , as labelled. ⇡ K v the behaviour of the relative -error as a function of the number of velocity grid-points L2 Nv, for various values of the Stokes number St and for a given scale r. One observes that the error decreases when the resolution increases, giving strong evidence of the convergence of the method. The error is found proportional to the velocity grid spacing ∆v, indicating 76 CHAPTER 4. LATTICE PARTICLES METHOD that the method is first order for large values of Nv. The constant is a decreasing function of the Stokes number. This indicates that the method is more accurate for particles with strong inertia. The reason for this trend will be addressed in the sequel. To assess the ability of the proposed method to reproduce physically relevant quan- tities, we now compare statistics obtained using the lattice method with those using a Lagrangian approach. We focus on the clustering and velocity difference properties that were introduced and discussed in section 4.4.1. Figure 4.8 shows for given values of the Stokes number and of the diffusivity, the second-order moment of the coarse-grained density (⇢E)2 as a function of r and various h r i values of the resolution in velocities, together with the value (⇢L)2 obtained with 106 h r i Lagrangian particles. One observes that the curves approach the limiting behaviour from below when the number of grid-points N becomes larger (i.e. when ∆v 0). At sufficiently v ! high velocity resolutions, the method is able to capture the large-scale properties of the concentration of the particles. The second-order moment of density then saturates to a value lower than that expected from Lagrangian measurements. The situation is very different at very low resolutions where the data obtained from the lattice-particle method deviates much, even at large scales. This corresponds to the case when the numerical diffusion in velocity is larger than the physical diffusion. These strong deviations stem from a non-trivial effect of diffusion that lead to finite- scale divergences of the solutions associated to different values of . In the absence of K diffusion, there is a finite probability that an order-one fraction of mass gets concentrated on an arbitrary small subdomain of the position-velocity phase space. This corresponds to a violent fluctuation where the local dimension approaches zero. At the time when this occurs, the mass distribution associated to a finite value of the diffusion will get stacked at a scale `d. Because of the chaotic nature of the particle dynamics, the two mass distributions, with and without diffusion will experience very different evolutions and diverge exponentially fast. Such a strong clustering event followed by the divergence of the solutions, is shown in Figure 4.9. Starting from a correctly reproduced distribution, the major part of non-diffusive Lagrangian particles concentrate into a subgrid region while the Eulerian approximation is stacked at scales of the order of `d. At a later time, the two distributions diverge and the diffusive particles fill faster larger scales. The probability with which one encounters such a configuration strongly depends on the Stokes number and on the spatial dimension. In the one-dimensional case, such events are rather frequent but become sparser when the Stokes number increases. This is essentially due to the compressibility of the carrier flow. For incompressible fluids in higher dimensions, we expect a negligible contribution from these events. In addition, we report some results on velocity difference statistics. For that, we have measured the first-order structure function S1(r) of the particle velocity, using (4.28) in the Lagrangian case and (4.29) for solutions obtained with the lattice-particle method. 6 Figure 4.10 shows the relative error of S1(r) for fixed values of the separation r =2⇡/2 = 25∆x, the Stokes number, and the diffusivity, as a function of the velocity resolution. 4.4. APPLICATION TO A ONE-DIMENSIONAL RANDOM FLOW 77

Figure 4.9: Three snapshots of the Lagrangian particles (black dots, for = 0) and of K the lattice-particle Eulerian solution (coloured background) in the (x, v) plane for different times: At tt? (Right), the Eulerian and Lagrangian solutions diverge exponentially fast with differences appearing at the largest scales..

Clearly, when the number of grid-points Nv increases, the error decreases, following a law approximatively proportional to the grid spacing ∆v. The inset shows the same quantity but, this time, for a fixed resolution (Nv = 33) and as a function of the Stokes number. One clearly observes a trend for this error to decrease with St. There are two explanations for this behaviour. First, as seen above, there are strong clustering events leading to differences between the Lagrangian and lattice solutions that can persist for a finite time. When the Stokes number increases, such events become less probable. The second explanation relies on the fact that particles with a larger Stokes number experience weaker velocity fluctuations. This implies that for a fixed value of Vmax, the particle velocity is more likely to be fully resolved at large values of St. As seen in the inset of Figure 4.10, the downtrend of the error is compatible with a behaviour St−1/2. It might thus be proportional to the / expected value of the root-mean-squared particle velocity when St 1 (see Abrahamson � (1975)), favouring the second explanation. To close this section on one-dimensional benchmarks of the lattice-particle method, we briefly assess the numerical cost of the method. The computational cost per time step for both Lagrangian and Lattice simulations are compared for a fixed resolution and coarse- 1 graining scale. A reference coarse-grained density field ⇢r from the Lagrangian method 7 NUM using a large number of particles (Np = 10 ) is compared to the ones ⇢r obtained either from the Lattice or the Lagrangian simulations varying the resolution (number of velocities 78 CHAPTER 4. LATTICE PARTICLES METHOD

102 100 L 1 /S 1 ) −1/2 L 10 1 St S L 1 − 10-1 /S E 1 ) S L 1 0 ( S 10

− 100 101 E 1 ∝ ∆v

S St (

10-1

10-2 101 102 Nv E Figure 4.10: Relative error between the particle velocity structure function S1 (r) obtained L from the lattice-particle method and that S1 (r) from Lagrangian averages, as a function of the number N of velocity grid-points. Here, the Stokes number is fixed St 1.9 and v ⇡ =2⇡ 10−3. Inset: same quantity but for N = 33 and as a function of the Stokes number K v St.

Nv for the lattice method or the number Np of Lagrangian particles). All the density fields are coarse-grained at the same scale r = L/128 as in Figure 4.8. Figure 4.11 shows the relative error as a function of the computational cost. All the simulations were done using the same quadri-core CPU with shared memory. We expect the costs to have the same tendencies when using distributed memory machines, as the communication should not vary much from one method to the other. For the Lagrangian simulations, the error is essentially given by finite number effects that affect the resolution of the p[article density −1/2 field. It is thus related to the statistical convergence of the average and behaves as Np . As the computational cost is proportional to Np, this explains the 1/2-scaling observed in 2 Figure 4.11. For the lattice simulations the cost is proportional to Nv , since increasing Nv with constant time step implies increasing Nx by the same factor. It appears that the Lattice method is more computationally efficient down to a given error. A few number of discrete velocities reproduces indeed better the reference density field than Lagrangian simulations with too few particles. The crossover observed in Figure 4.11 is due to the scaling of the error with respect to Nv that is slightly slower than linear, as shown in Figure 4.8. Note that if the advection term in equation (4.14) is treated with a higher order scheme, the error as a function of Nv will decrease faster than linearly, and the Lattice method will be computationally advantageous if lesser errors are targeted. Finally, let us stress that the comparison is here performed for the simple case of non interacting particles. In some situations involving long-range interactions (such as gravitational or 2 electrical forces), Lagrangian methods might require O(Np ) operations and become much less efficient that the lattice method, which will then still have a cost O(N N ). x ⇥ v 4.5. APPLICATION TO INCOMPRESSIBLE TWO-DIMENSIONAL FLOWS 79

100 Lattice Lagrangian − 1 2 slope ⟩ ∞ r ρ ⟨

/ -1

⟩ 10 ∞ r ρ − NUM r ρ ⟨ 10-2

10-5 10-4 10-3 10-2 10-1 Computational cost (sec / time step)

Figure 4.11: Relative errors on the coarse-grained density fields obtained with either L2 Lagrangian or lattice simulations, plotted here as a function of the computational cost (same parameters as in Figure 4.8 for St =2.5). The various simulations were done with a fixed time stepping ∆t =5 10−2; the lattice case spans N = 3 to 65, while the Lagrangian ⇥ v simulations correspond to a number of particles N varying from 103 to 2 106. The symbol p ⇥ in the y-axis label stands for either Lagrangian of Eulerian simulations.

4.5 Application to incompressible two-dimensional flows

Numerical integration in more than one dimension consists of the same routines as in Section 4.3, each operator being applied on one dimension after the other.

4.5.1 Cellular flow We first consider a fluid flow that is a stationary solution to the incompressible Euler equations (and to the forced Navier–Stokes equations). It consists of a cellular flow field, a model that have often been used to investigate mixing properties, as well as the set- tling of heavy inertial particles (see, e.g., Maxey (1987); Bergougnoux et al. (2014)). The velocity field is the orthogonal gradient of the L-periodic bimodal stream function (x, y)=U sin(⇡(x + y)/L) sin(⇡(x y)/L) (the typical velocity strength is here denoted − by U). The cellular flow has been here tilted by an angle ⇡/4 in order to avoid any align- ment of the separatrices between cells with the lattice that leads to spurious anisotropic effects. Figure 4.12 shows two snapshots for two different values of St = ⌧pU/L of the sta- tionary particle distribution (black dots), together with the density field evolved by the lattice-particle dynamics. For the smallest Stokes number (Left panel), one observes that the particle distribution is concentrated along the separatrices between the different cells. One also observes that it develops entangled structures in the vicinity of the hyperbolic 80 CHAPTER 4. LATTICE PARTICLES METHOD

Figure 4.12: Particles stationary distribution inside a tilted cellular flow along with the density field from the lattice method. The value of the diffusivity is =8⇡ 10−3. Left: K ⇥ St =1/(2⇡). Right: St =1/⇡. These simulations were performed on a lattice with 10242 spatial grid-points associated to 192 discrete velocities. The distributions have been here spatially shifted in order to avoid having the concentration point (0, 0) at the origin. stagnation points of the flow. These loops, which are aligned with the stable direction, cor- responds to oscillations in the particle dynamics that occurs when their inertia makes them cross the unstable manifold with a too large velocity. At larger St, the particle distribution is somewhat broader but is this time centred on specific trajectories that do not perform the aforementioned oscillation but rather cross ballistically the heteroclinic separatrices. In both cases the particle distribution contains regions where trajectories are clearly crossing each other. The lattice method reproduces fairly well this complex dynamics. Note that any traditional Eulerian-Eulerian method introducing a particle velocity field will not be able to reproduce such effects. One may wonder why the flow has been tilted by an angle. Figure 4.13 (left) shows the density field obtained for a simulation with a non-tilted flow for St =1/⇡. Unfortunately, it is unable to recover the correct particle spatial repartition, shown on the right panel by the Lagrangian particles (black dots). Actually, mass has concentrated along the steady, stable manifolds between the rotating cells. At each passage through the hyperbolic points (cell corners), for which u = 0, a little amount of mass is transferred into the corresponding cell in velocity space [ ∆v/2, ∆v/2]2, from which no mass ever escape, due to the stationarity − of the velocity field. Trying to circumvent this effect, a refinement of the finite volume scheme was attempted, adding an outgoing flux proportional to the difference between u(x) v , with v the centre of the cell. This ensured an ejection even for the cell for − i i which vi = 0. But it only spread the distribution a little along the concentrating, stable manifold. This effect is expected to be present for all finite ∆v. This presently limits the capacity of the model to represent correctly flows with stationary manifolds aligned with 4.5. APPLICATION TO INCOMPRESSIBLE TWO-DIMENSIONAL FLOWS 81

Figure 4.13: Illustration of the particle density integrated by the lattice method in the case of a non-tilted cellular flow. St =1/⇡ and =8⇡ 10−3. This picture is to be compared K ⇥ with Figure 4.12 (right).

one of the lattice propagation direction.

4.5.2 Heavy particles in 2D turbulence

We next turn to the study of the model in non-stationary fluid flows that are solutions to the forced two-dimensional incompressible Navier–Stokes equations (2.1). The fluid velocity field u is computed numerically using a pseudo-spectral, fully de- aliased GPU solver for the vorticity streamfunction formulation of the Navier–Stokes equa- tion (2.18). The two-dimensional Navier–Stokes equation is known to develop two cascades, as explained in Section 2.2.2. Dimensional analysis predicts that the direct enstrophy cascade is associated to a unique −1/3 timescale ⌧Ω = ✏! . Investigating heavy particle dynamics at the small scales of two- dimensional turbulence thus requires comparing their response time to ⌧Ω. The relevant parameter is then the Stokes number defined as St = ⌧ /⌧ . For St 1, particles almost p Ω ⌧ follow the flow and tend to distribute homogeneously in space. When St 1, they � completely detach from the fluid and experience a ballistic motion leading again to a space- filling distribution. Non-trivial clustering effects occur when the Stokes number is order one. This is illustrated in Figure 4.14, which shows a snapshot of the particle distribution in the position space on top of the turbulent vorticity field in the direct enstrophy cascade. Due to their inertia, particles are ejected from vortices and concentrate in high-strain regions. 82 CHAPTER 4. LATTICE PARTICLES METHOD

2

1.8

1.6

1.4

1.2 D2 1 ζ1 0.8

0.6

0.4

0.2

0 0 0.2 0.4 0.6 0.8 1 St

Figure 4.14: Left: Snapshot of the position of particles (black dots) for St =0.1. The coloured background shows the vorticity field obtained from a 10242 direct numerical sim- ulation with a large-scale forcing at wavenumbers 1 k < 4. Right: Correlation dimen-  | | sion and scaling exponent ⇣ of the particle velocity first-order structure function as a D2 1 function of St in the two-dimensional direct enstrophy cascade.

There, the combination of stretching, folding and dissipation induced by their dynamics makes them converge to a dynamical attractor with fractal properties. Such a behaviour is quantitatively measured by the correlation dimension defined in equation (4.26). The D2 evaluation of as a function of St resulting from Lagrangian simulations is presented in D2 Figure 4.14. At St = 0, unlike in the one-dimensional case where the dimension of the attractor is 0, particles follow the streamlines of the incompressible two-dimensional flow, fill the position space, and hence = 2. Clustering then increases with inertia to attain D2 a minimum at St 0.2. It then decreases again as the velocity of particles separate from ⇡ that of the fluid and disperse in the velocity space, leading to a space-filling distribution = 2 when St . D2 !1 The velocity distribution of particles is itself having a behaviour that is very similar to the one-dimensional case. This is clear from Figure 4.14, where the scaling exponent ⇣1 of its first-order structure function (see equation (4.28)) is represented as a function of the Stokes number. For St 1, the particles are as if advected by a smooth velocity field and ⌧ ⇣ 1. When St & 1, particles with very different velocities can come arbitrarily close to 1 ⇡ each other and ⇣ 0. 1 ! Particle properties in the inverse energy cascade are more difficult to characterize be- cause of the scale-invariance of the velocity field. In particular, neither the moments of the coarse-grained density nor the particle velocity structure functions display any scaling behaviour. What has been nevertheless observed numerically by Boffetta et al. (2004) is that the particle spatial distribution is dominated by the presence of voids whose sizes 4.5. APPLICATION TO INCOMPRESSIBLE TWO-DIMENSIONAL FLOWS 83 obey a universal scaling law. Chen et al. (2006a) argued that such voids are related to the excited regions of the flow and that particles tend to follow the calm regions where the zeros of the fluid acceleration are more probable. In the sequel we apply the lattice method to both the direct and the inverse two- dimensional cascades. Resolving both cascades in the same simulation would require a tremendous scale separation and thus number of grid-points (see Boffetta & Musacchio (2010)). For that reason we consider the two cases separately.

Direct enstrophy cascade The fluid flow is integrated by a pseudo-spectral method on a uniform square spatial grid using a streamfunction formulation of the Navier–Stokes equation (2.18). To maintain a developed turbulent state, a stochastic forcing is applied in the wavenumber shell 1  k < 4 of Fourier space while the kinetic energy accumulating at large scales is removed | | by a linear friction. The particle dynamics is simulated using a spatial lattice with the 2 same resolution as the fluid and with various numbers Nv of discrete velocities. The acceleration step is done via a flux limiter scheme as described in Section 4.3. Results are compared to particle trajectories obtained from Lagrangian simulations. Figure 4.15 shows the instantaneous particle distributions obtained from the two approaches. The qualitative agreement is excellent, reproducing correctly depleted zone as well as more concentrated regions.

Figure 4.15: Snapshot of the position of Lagrangian particles (black dots), together with the density field obtained from the lattice method (coloured background, from white: low densities to red: high densities) for St 0.1 at the same instant of time as Figure 4.14. The ⇡ fluid flow was integrated using a resolution of 10242 and lattice simulations were performed with 10242 spatial grid-points associated to 172 discrete velocities. 84 CHAPTER 4. LATTICE PARTICLES METHOD

To get more quantitative informations on the relevance of the method, we have per- formed a set of simulations with a 5122 resolution and in which both the number of discrete 2 velocities Nv and the maximum velocity Vmax are varied. Figure 4.16 shows measurements of the second-order moment of the coarse-grained density ⇢r obtained by integrating the phase-space density f(x, v,t) with respect to velocities and over space in boxes of length r. This is the two-dimensional version of Figure 4.8 and the statistics of ⇢2 have a very h ri similar behaviour as in the one-dimensional case. Here St =0.5, Vmax =3.9 urms, and Nv is varied from 9 to 21. One clearly observes that the statistics obtained from the lattice method converges to that obtained from Lagrangian simulations.

-1 1.5 10 1.45 -2 1.4 E 10 1.35 -3 1.3 10

r 1 2 3 45 ⟩ 2 1.25 Vmax/urms ρ ⟨ 1.2 10-1 1.15 Nv =9 Nv = 13 Nv =9 -2

1.1 Nv = 17 E 10 Nv = 13 N = 21 N = 17 1.05 v v Lagrangian Nv = 21 1 10-3 10-3 10-2 10-1 100 0.2 0.4 0.6 0.8 1 1.2 1.4 r/L ∆v/urms

Figure 4.16: Left: Second-order moment of the coarse-grained density ⇢r as a function of r for St =0.5 and =1.6⇡ 10−1 in the direct cascade, both from Lagrangian measurement K (black line) and the lattice method with different Nv, as labeled. Right: Distance-averaged error of the second-moment of the mass density as defined in (4.31) as a function of E the maximal velocity V (top) and of the velocity grid spacing ∆v =2V /(N 1) max max v − (bottom) for various values of the velocity resolution N 2 and for St 0.5. v ⇡

The interplay between the choices of Nv and of Vmax requires some further comments. On the one hand the method converges when both ∆v =2V /N 0 and V . max v ! max !1 On the other hand, the computational cost is N 2. One can thus wonder if for a fixed / v cost there is an optimal choice of Vmax that minimizes the error obtained with the lattice method. Focusing again on the second-order statistics of the particle mass distribution, we have measured the average with respect to r of the error made on the density moment ⇢2 h ri defined as (N ,V )= (⇢E)2 (⇢L)2 / (⇢L)2 , (4.31) E v max h r i−h r i h r i where 1 L f(r)= f(r) r dr. (4.32) L2 | | Z0 4.5. APPLICATION TO INCOMPRESSIBLE TWO-DIMENSIONAL FLOWS 85

Figure 4.16 (top) represents this quantity as a function of Vmax for different values of 2 the cost Nv . One clearly observes that there is indeed for a fixed Nv a specific choice of Vmax where the error is minimal. The optimal value of the maximal velocity increases with Nv. On the right of the minimum, the error is in principle dominated by a ∆v too large. This is confirmed by the collapse of the various curves on the right of their minima that can be seen in the bottom of Figure 4.16 where is represented as a function of ∆v. In the E left of the minimum, the error should be dominated by a too small value of Vmax. One can indeed guess an asymptotic collapse for V u on the upper panel of Figure 4.16, or max ⌧ rms equivalently, the fact that the curves separate from each other at small values of ∆v in the lower panel.

The value of the error at the optimal Vmax decreases from Nv = 9 to Nv = 13 but then seems to saturate (or to decrease only very slowly) at higher values of Nv. One cannot exclude that this behaviour corresponds to a logarithmic convergence of when N . E v !1 This slow dependence is also visible in the bottom panel where the collapse of the various curves seems to extend weakly on the left-hand side of the minima for Nv = 13, 17 and 21. Accordingly, a small difference in Nv is not enough to decrease significantly the error. In the specific case considered (for St 0.5 in the direct cascade), the resulting optimal ⇡ −3 choice seems to be Nv = 13 with Vmax =2.25 urms, which leads to a relative error 10 .

Inverse energy cascade

To complete this study we have also tested the proposed lattice method in a two-dimensional turbulent flow in the inverse kinetic energy cascade regime. The stochastic forcing is now acting at small scales (400 k 405) and we made use of hyper-viscosity (here eighth  | |  power of the Laplacian) in order to truncate the direct enstrophy cascade. The kinetic energy accumulated at large scales is again removed using a linear friction in the Navier– Stokes equation (2.18). The particle Stokes number is now defined as St = ⌧p/⌧L using the large-eddy turnover time ⌧L = L/urms since small-scale statistics are dominated by forcing and are thus irrelevant. The flow is integrated with a resolution of 20482 grid-points while the lattice-particle method is applied for St 0.1 on a coarser grid with 5122 points. ⇡ Figure 4.17 shows that the lattice-particle method is able to reproduce the main qual- itative features of the particle spatial distribution at scales within the inertial range of the inverse energy cascade. This is confirmed in Figure 4.17 which represents the relative error defined in equation (4.31) of the second-order moment ⇢2 of the density ⇢ coarse- E h ri r grained over a scale r. The Lagrangian integration was performed with 2 107 particles ff 2 ⇥ 2 2 (with no physical di usion) and the lattice method on a 512 spatial grid with Nv =9 discrete values of the particle velocity. One clearly observes that the error decreases at the largest scales of the flow. 86 CHAPTER 4. LATTICE PARTICLES METHOD

100

10-1 E

10-2

10-3 10-3 10-2 10-1 100 r/L

Figure 4.17: Left: Snapshot of the position of Lagrangian particles (black dots), together with the density field obtained from the lattice method (coloured background, from white: low densities to red: high densities) for St 0.1 in the inverse energy cascade. The fluid ⇡ flow was integrated using a resolution of 20482 and lattice simulations were performed with 5122 spatial grid-points associated to 92 discrete velocities. Right: Relative error of E the second-order moment of the coarse-grained density ⇢r as a function of r for St =0.5. The lattice-particle method was here used with 5122 position grid-points and 92 velocity grid-points.

4.6 Conclusions

We have presented a new Eulerian numerical method to simulate the dynamics of inertial particles suspended in unsteady flow. This lattice-particle method is based on the dis- cretisation in the position-velocity phase space of the evolution equation for the particle distribution. The spatial grid is chosen such that particles with a given discrete velocity hop by an integer number of gridpoints during one time step, an idea close to that used in lattice-Boltzmann schemes. We have shown that the method reproduces the correct dynamical and statistical properties of the particles, even with a reasonably small amount of velocity gridpoints. Some deviations from Lagrangian measurements are nevertheless observed at small scales in one dimension. We obtained evidence that they are due to nu- merical diffusivity acting in the space of velocities and are more important in one dimension at small Stokes numbers than otherwise. The proposed method is anyway intended to de- scribe large scales where such deviations disappear. It might hence be a suitable candidate for developing large-eddy models for particle dynamics. Indeed, as equation (4.19) is linear in f, some techniques of subgrid modeling used in scalar turbulent transport (see Girimaji & Zhou (1996) for example) could be adapted. Our approach consists in always imposing the same mesh for particle velocities, in- 4.6. CONCLUSIONS 87 dependently of the spatial position and of the local value of the fluid velocity. This is particularly well-adapted for particles with a large Stokes number. Their velocity experi- ences small fluctuations and is generally poorly connected to that of the fluid. In addition, the method is accurate at the largest scales and can hence catch the structures appearing in the spatial and velocity distributions of large-Stokes-number particles. Such considerations indicate that the proposed lattice-particle method is suitable for simulating particles with a sufficiently strong inertia. Conversely, particles with a weak inertia develop fine-scale structures in their distribution. They result from tiny departures of their velocity from that of the fluid. Our method, applied with a fixed velocity resolution, might not be able to catch such deviations. However, a more suitable idea for this case is to use a variation of our approach where, instead of a full resolution of the particle velocity, one considers its difference with that of the fluid. This would of course require changing scheme for integrating advection but should in principle not lead to any major difficulty. 88 CHAPTER 4. LATTICE PARTICLES METHOD CHAPTER 5

Turbulence modulation by small heavy particles

In this chapter, turbulence modulation by small heavy particles is studied in the two- dimensional enstrophy cascade. The two asymptotics of small and high inertia are consid- ered. In both of them, we make use of a fluid description for the solid suspension. This allows us to treat two-way coupling in an exact manner. The net force exerted by the particles on the fluid phase is computed exactly as the result of the action-reaction rule from the number density function. In the small Stokes regime, the bi-fluid approximation is used, yielding an evolution equation for the density and the velocity of the solid phase. It is shown that particles ejected from vortices core develop a shear instability at their boundaries. This results in a decrease of the slope of the fluid kinetic energy spectrum, together with an enstrophy injection. This last effects results in the reversal of the sign of the nonlinear flux. This study has made the object of a paper submitted to the journal Physical Review Letters which is reproduced hereafter. In the large Stokes asymptotics, the lattice particle method presented in the previous chapter was coupled to the fluid phase through the particle number density function. The effect of two-way coupling in this case was shown to behave as an additional effective large- scale friction. As the inertia was decreased, the qualitative behaviour observed in the limits of small Stokes numbers (see above) began to be recovered. This study has made the object of a paper submitted to the journal Journal of Fluid Mechanics which is reproduced hereafter.

89 This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics 1

Modulation of two-dimensional turbulence by small heavy particles

Fran¸coisLaenen,1 Stefano Musacchio,2 and J´er´emieBec1 1Universit´eCˆoted’Azur, Observatoire de la Cˆoted’Azur, CNRS, Laboratoire J.-L. Lagrange, 06300 Nice, France. 2Universit´eCˆoted’Azur, CNRS, Laboratoire J.-A. Dieudonn´e, 06108 Nice, France

(Received xx; revised xx; accepted xx)

The question of two-way coupling, that is of the back effects of a transported particle phase onto the fluid flow, is adressed in this paper. The suspension considered is that of small-Reynolds-number heavy particles with Stokes numbers of the order of unity or larger, that are embedded in a two-dimensional turbulent flow in the direct enstrophy cascade. A recently developed numerical approach providing a description of the particle in terms of fields is used. This method is based on the kinetic equation for the position- velocity phase-space distribution of the particles and allows for an exact expression of the back-reaction from the particles from the action-reaction principle. It is shown that global quantities such as total energy and enstrophy monotonically decrease with the mass ratio, as well as Ekman and molecular dissipation. At large Stokes numbers, the effect of particles is explained in terms of an effective additional Ekman friction. This has the consequence to steepen the power-law behavior of the enstrophy spectrum. The impact on the intermittent properties are also examined. In particular, it is shown that the multiscaling properties of the vorticity structure function follow a simple renormalisation given by second-order statistics. At scales within the dissipative range, this renormalisation stops working. The probability distribution function of the vorticity gradients display tails which broaden when the coupling intensity is increased. Finally, it is also shown that two-way coupling enhances particle clustering at large Stokes numbers. This effect can be attributed to a decrease of the actual particles inertia, making them more likely to be ejected from coherent vortices and to form preferential concentrations.

Key words: turbulence modulation, particle-laden flows, turbulent flows, multiphase flows modelling

1. Introduction Numerous natural and industrial situations involve particle-laden flows. These include pollutant transport (Gyr & Rys 2013), plankton dynamics and accumulation (Durham et al. 2013), the modelling of fluidized beds in pharmaceutical or mining manufacturing processes (Curtis & Van Wachem 2004) and dilute spray combustion (Li et al. 2012). Finite-inertia particles display the important feature of detaching from the flow stream- lines. Those heavier than the surrounding fluid (such as dust or liquid droplets in gases) tend to be ejected from vortices by centrifugal forces and form preferential concentrations in strain-dominated regions. This effect is expected to enhance turbulent collisions and, as a matter of fact, to trigger coalescences between droplets and accelerate the initiation 2 F. Laenen, S. Musacchio, J. Bec of rain in warm clouds (Falkovich et al. 2002; Lanotte et al. 2009). During this small- scale process, particle with very different histories can possibly meet to collide, bringing together different turbulent velocities (Gustavsson et al. 2008) or non-trivial inertial- range correlations of transport (Bec et al. 2016). Consequently, this microphysical effect is strongly coupled to the full range of length and timescales that are excited by turbulent fluctuations, leading to significant difficulties in establishing reliable quantitative models. The back reaction exerted by the particles onto the fluid gives rise to an extra complexity in modelling. Such a two-way coupling typically arises when the mass fraction of the dispersed particle phase relative to the carrier fluid is sufficient to provoke momentum exchanges with the surrounding medium (Elghobashi 1994), resulting in a modulation of the turbulent flow. The understanding of the mechanisms at play, even at a purely qualitative level, is still insufficient for developing efficient models of relevance to applications. An instance where turbulence modulation plays a key role is in the process of planet formation during the early stages of the Solar system. It has indeed been shown that the interactions between dust and gas in protoplanetary disks create an instability facilitating grain clumping (Johansen & Youdin 2007; Jacquet et al. 2011) and we are still lacking quantitative predictions on how much this enhances the formation of planetesimals. Many questions on the modelling of turbulence modulation by particles remain also open in engineering applications. These include turbulent sprays (Jenny et al. 2012) or fuel droplets in combustion chambers (Post & Abraham 2002), where two-way coupling is expected to enhance heat transfers and macroscopic chemical reaction rates. The development of a comprehensive phenomenology and of efficient models is hindered by the complexity of the problem. A source of complexity comes from the possibility or not to model the particle dynamics itself. For instance, when their sizes are comparable or larger than the smallest active scale of the fluid flow (e.g. the Kolmogorov dissipative scale in three-dimensional turbulence), determining particles dynamics requires fully resolving the fluid flow around them and integrating viscous strain and pressure at their surface to obtain net forces. Local modulation of the flow around the particles might then be assessed, as well as modification of global quantities. Various numerical techniques have been developed to the end of studying the effects induced by such finite-size particles. Immersed boundaries (Lucci et al. 2010; Cisse et al. 2013), two-fluid level-set (Sabelnikov et al. 2014; Loisy & Naso 2016) and Lattice-Boltzmann methods have already been used in this context (Poesio et al. 2006; Gao et al. 2013), allowing one to reach volume fractions of the order of 2-10 % (see, e.g., Ten Cate et al. 2004). However, the number of resolved particles is generally limited because of the high computational demand and the effects on turbulence do not manifest in a sufficiently intelligible manner. Experiments have been conducted in order to span higher values of the dispersed phase volume fraction. It was found that in wall flows, the presence of particles can either trigger or prevent the transition to turbulence depending on the size and volume fraction of the particles (Matas et al. 2003). In settings closer to homogeneous isotropic turbulence (as, e.g., in a von Karman flow), it was observed that large neutrally buoyant particles attenuate turbulent fluctuations without affecting their nature: The distribution of acceleration statistics and the velocity structure functions are only affected by the modifications of the Reynolds number (Cisse et al. 2015). In addition, these measurements suggest a mechanism of particle shedding that prevent turbulent fluctuations from reaching the center of the experiment, advocating that the attenuation of turbulence might strongly depend on the particle size, on its mass, and on the way kinetic energy is injected in the flow. On this account, and as stressed in Saber et al. (2015), important efforts are still needed in order to build up a phenomenological outlook and to understand whether or not any universal ideas and concepts can be sketched for the modulation of turbulence by particles. 3 Such difficulties put forward the need to investigate in more details simpler and more easily controllable situations. A straightforward choice is to focus on very small heavy spherical particles whose dynamics is fully determined by a unique parameter, the Stokes number St. Experimental and numerical studies suggest that the effect of small particles is to attenuate turbulence intensity at low inertia, i.e. St . 1 or low volume fractions, and to enhance it when St & 1 and inertia effects become large (Mandø 2009; Patro & Dash 2014). Furthermore, modulation is not necessarily isotropic nor homogeneous in space. Gualtieri et al. (2011) performed simulations with particles with St of order one and that are suspended in a flow with mean shear. The effect of particles exhibits a strong anisotropy: The kinetic energy spectrum displays an excitation of the small scales that clearly indicates that two-way coupling is responsible for a transfer of large-scale anisotropies to the small scales at the expense of the inertial-range energy content. Yang & Shy (2005) investigated intermediate Stokes number (0 < St < 2) and computed a wavelet-based energy spectrum. They observed an increase of the turbulent fluctuations at high frequencies, maximal at St ⇠ 1, and more pronounced in the direction of gravity. Hwang & Eaton (2006) considered suspensions with a larger value of the Stokes number (St ⇠ 50) and showed a decrease of the total kinetic energy with increasing mass load. They observed a uniform attenuation across the scales in the horizontal plane and a dissipation less pronounced in the vertical direction. An advantage of considering small particles is that they are generally associated with a low Reynolds number, so that the forces exerted on them by the fluid can be expressed explicitly. While this could seems as a strong point to perform efficient numerical simulations, it actually leads to other difficulties. In principle, considering small particles foster the use of Euler-Lagrange numerical approaches and thus of a point-force method for the back reaction on the flow. This method originates from the particle-in-cell method (Crowe et al. 1977) and consists in considering particles as punctual sources and sinks of momentum. The total forces exerted on the fluid phase by all the particles in a given mesh cell must then be distributed to the neighbouring grid points. As a result, this method is strongly grid-dependent (Balachandar 2009). It greatly relies on the number of particles in each cell, and can suffers from a lack of numerical convergence when this number is too low (Garg et al. 2009). This effect can be particularly penalizing in the case of particles with inertia that exhibit preferential concentration and distribute in space in a very non-uniform manner. Progress are however made in order to improve Euler-Lagrange approaches. For instance, Gualtieri et al. (2015) used physical arguments to propose a specific regularisation method that is expected to remove most drawbacks linked to the extrapolation of the point-force to the fluid gridpoints. Despite such developments, Euler-Lagrange methods might still not fit situations requiring by essence a large number of particles. These include of course situations where particles are not sufficiently dilute but also cover the case of particles with large Stokes numbers. Such particles are indeed known to develop a broad dispersion in velocity and a grounded representation requires sufficiently many particles to map the full position-velocity phase-space. The objective of the work reported here is twofold. The first target is to concentrate on a case amenable to a systematic analysis, namely the modulation of two-dimensional incompressible turbulence by small heavy particles. We focus on the direct cascade, where enstrophy (squared vorticity) injected at the large scales, is transfered through the scales, before being dissipated by molecular viscosity (see Boffetta & Ecke 2012, for a review). Besides its relevance to horizontal, large-scale geophysical motions (see, e.g., Danilov & Gurarie 2000), two-dimensional turbulence profits from being easy to simulate numerically. On the one hand, this allows for broadly exploring the dependence upon the two parameters which characterise the coupling with particles, namely their 4 F. Laenen, S. Musacchio, J. Bec

Stokes number St and their total mass fraction φm. On the other hand, undemanding simulations enable integrations over very long periods, typically of several hundreds of large-eddy turnover times, a necessary duration to guarantee the convergence of large- scale quantities and to gather accurate statistics. The other goal of this work is to propose and exploit a new Eulerian numerical technique that copes with the difficulties encountered when using Lagrangian point-force methods and allows for considering the effect of particles with moderately large Stokes numbers. We make use of the method introduced in Laenen et al. (2016), which integrates explicitly the kinetic equation for the particle population without introducing any approximation on the dynamics. The paper is organised as follows. We formulate the dynamical model in §2 and describe the numerical method in §3. Next, we describe in §4 the effects of two-way coupling on the statistics of the fluid energy and enstrophy. We demonstrate that the presence of large-Stokes-number particles leads to decrease both the dissipation rate and the global values of energy and enstrophy, and concomitantly steepen the corresponding power spectra. Conversely, the feedback of particles with St ⇠ 1 causes an increase of enstrophy dissipation and is responsible for an injection of energy at intermediate scales. In §5, we then present results on small-scale, higher-order statistics. We show that intermittency increases with the particle mass loading. In particular, the probability density functions of the vorticity gradients are found to develop larger tails when the coupling with particles is strengthened. In §6 we report measurements on particle clustering and find that the coupling can either enhance or deplete it depending on the value of the Stokes number. We finally draw some concluding remarks in §7

2. Phase-space description As stated in the Introduction, we adopt in this work a description of particle suspen- sions in terms of fields. This method does not rely on approximating the particle dynamics in terms of a particle velocity field but considers the full phase-space distribution f(x, w,t) of particles positions and velocities. This quantity is defined as

N 1 p f(x, w,t)= δ(X (t) − x) δ(V (t) − w), (2.1) N i i p Xi=1 where Np is the total number of particles in the domain, Xi and Vi are the position and the velocity, respectively, of the i-th particle. Note that the phase-space distribution f is normalized in such a way that its integral over positions and velocities is equal to unity. The evolution of f is governed by the Liouville equation, which expresses conservation in phase space, namely

∂tf + rx · (w f)+rw · (Ff!p f) = 0 (2.2) where Ff!p(x, w,t) denotes the sum of the forces acting at time t on a particle at position x with velocity w. These various forces can be very complex when including all possible effects, such as buoyancy, Basset–Boussinesq history forces due to the interaction of the particle with its own wake, the added-mass factor, Fax´encorrections, etc. (see Maxey & Riley 1983, for details). We focus here on the case of small, very heavy, low-Reynolds- number spherical particles. Namely, we assume that the particles radius a is much smaller than any active scale of the flow and that their material mass density ρp is much larger than the fluid density ρf and or the sake of simplicity we also neglect the effect of gravity. 5 Under these assumptions, the forces reduce to the Stokes drag: 1 Ff!p(x, w,t)=− (w − u(x,t)) , (2.3) τp where u(x,t) denotes the fluid velocity evaluated at the particle position and τp is the 2 particle response time given by τp =2a ρp/(9 ρf ν), with ν the fluid kinematic viscosity. The fluid velocity field u(x,t) evolves according to the two-dimensional incompressible Navier-Stokes equations 2 ∂tu +(u · rx) u = −rxp − α u + νrxu + Fu + Fp!f , rx · u =0. (2.4)

The flow is sustained in a stationary turbulent regime by the external Gaussian force Fu which is assumed to be concentrated on the large spatial scales and white noise in time. The linear damping −α u represents Ekman friction and prevents the pile-up of kinetic energy at large scales. The force Fp!f exerted by the particles onto the fluid follows from the action-reaction principle and reads

2 F ! (x,t)=−φ F (x, w,t) f(x, w,t)d w (2.5) p f m Z p

This volume forcing depends on the mass ratio parameter φm = Np Vp ρp/(V ρf ), where Vp is the volume of a single particle, Np is the total number of particles in the domain whose volume is denoted Vf . As we consider very heavy particles (ρp/ρf � 1), one can possibly get finite values of φm even if the volume fraction NpVp/V is small. We further introduce the particle density np and the average particle velocity vp as

n (x,t)= f(x, w,t)d2w, (2.6) p Z

n (x,t) v (x,t)= w f(x, w,t)d2w (2.7) p p Z

Note that np is not a mass density field. Because of the definition of f, it actually counts the number of particles (and is rather a numer density) but it is normalised as 2 np(x,t)dx = 1. It is worth keeping in mind that for particles with large Stokes numbers,R the actual velocities at a given spatial position can be very dispersed and generally differ from the average velocity field vp. Using (2.6), (2.7), and the Stokes drag expression (2.3), one obtains the following expression for the force exerted by the particles onto the fluid

φm Fp!f (x,t)=− np(x,t)[u(x,t) − vp(x,t)] . (2.8) τp Usually, the dynamics of incompressible two-dimensional flows is formulated in terms of the streamfunction ψ(x,t), such that u =(∂2ψ, −∂1ψ), the indices 1, 2 being associated to the two spatial dimensions, and of the scalar vorticity ω = rx ⇥ u = ∂1u2 − ∂2u1 = 2 −rxψ. The vorticity-streamfunction formulation is obtained from the curl of Navier– Stokes equation ((2.4)) and reads

2 φm ∂tω + J (ψ, ω)=νrxω − αω+ Fω − [np rx ⇥ (u − vp)+(u − vp) · rxnp] , (2.9) τp where we introduced the two-dimensional Jacobian J (ψ, ω)=∂1ψ∂2ω − ∂2ψ∂1ω and Fω = rx ⇥ Fu. In the sequel we investigate the modification by particles of a fundamental turbulent state. This primary regime is obtained by setting φm = 0 and fixing the characteristics of 6 F. Laenen, S. Musacchio, J. Bec

+Wmax w u(x,t) f(x, w,t) Velocity w ∆ −Wmax ∆x Position x Figure 1. Sketch of the algorithm used to update the position-velocity phase-space particle distribution f(x, w,t). The positions are discretized with a resolution ∆x and the velocities, bounded between −Wmax and +Wmax with a resolution ∆w. At each time step,the dynamics is split in two operations: advection with the corresponding velocity (horizontal arrows) and relaxation to the fluid velocity (vertical arrows).

the external forcing Fu and the values of the dissipative constants ν and α. It is chosen to correspond to a developed direct enstrophy cascade from which a root-mean-squared vorticity ωrms can be measured. This value, obtained with no coupling to the particle phase, serves as a reference. Deviations due to the coupling between the particle and the fluid phases are obtained by varying only two control parameters: the non-dimensional mass loading φm defined above and the Stokes number St 0 = τp ωrms obtained by non-dimensionalising the particle response time by the root-mean-squared value of the vorticity obtained with no coupling.

3. Numerical method The incompressible fluid flow is integrated numerically using a spectral, fully de- aliased solver based on the vorticity-streamfunction formulation (2.9) of the Navier- Stokes equations. Time-marching is done using a second-order Runge–Kutta scheme. This equation is coupled to the time evolution of the particle phase-space distribution f given by the Liouville equation (2.2). This integration is performed by means of the lattice-particle method introduced by Laenen et al. (2016) and which has been shown to reproduce well the concentration and velocity properties of inertial particles in random and turbulent flows. The lattice-particle algorithm is inspired from the lattice-Boltzmann method (see, e.g., Succi 2001, for a review). The idea is to approximate the distribution f as a piecewise 2 2 constant field discretized in phase-space on a square lattice with resolution Nx ⇥ Nw. For a square periodic spatial domain of size L, the resolution in particle positions is 2 ∆x = L/Nx. The velocity domain is the bounded square [−Wmax, +Wmax] and is divided in squares of size ∆w =2Wmax/Nw. The time step size is chosen such that ∆x = ∆w ∆t. The algorithm to update f between two consecutive time steps consists in two successive operations (illustrated in Fig. 1 in the one-dimensional case for simplicity). The first step consists in advecting on the spatial domain the mass situated in the cell x =(i1 ∆x, i2 ∆x) with velocity w =(j1∆w, j2∆w). This mass is displaced to the cell 0 x = ((i1 + j1) ∆x, (i2 + j2) ∆x) associated to the same velocity w. The second step consists in updating the value of f according to the conservation law (2.2) to account for the relaxation of the particle velocity to that of the fluid at the same location. This is performed using a finite-volume scheme in the velocity direction, which is based on 7

φm τp EZηC ην ηα

00.50.97 2.1 00.047 0.021 0.10.50.43 1.60.033 0.019 0.016 0.20.50.34 1.40.038 0.015 0.014 0.30.50.29 1.30.039 0.016 0.013 0.40.50.22 1.20.038 0.018 0.012 0.50.50.19 1.10.036 0.019 0.011 0 21.10 2.4 00.047 0.024 0.1 20.40 1.80.033 0.021 0.018 0.2 20.26 1.10.046 0.011 0.011 0.3 20.27 1.20.05 0.0083 0.012 0.4 20.19 0.88 0.054 0.0055 0.0088 0.5 20.16 0.83 0.057 0.0049 0.0083 0 81.10 2.3 00.047 0.023 0.1 80.38 1.30.021 0.032 0.013 0.2 80.30 1.20.033 0.020 0.012 0.3 80.23 1.00.044 0.016 0.010 0.4 80.19 0.89 0.047 0.0097 0.0089 0.5 80.17 0.83 0.051 0.0072 0.0083

Table 1. Parameters used for the numerical simulations. φm is the particle mass fraction, τp their response time, E is the measured average kinetic energy of the fluid and Z its average enstrophy. ηC is the average enstrophy dissipation due to the coupling, ην the average enstrophy dissipation due to viscosity and ηα the average enstrophy dissipation due to Ekman friction. In all cases, the spatial resolution, i.e. the number of spatial collocation points for the fluid phase 2 2 and the number of spatial cells for the solid phase, is always equal to Nx = 1024 . The number 2 2 of velocity cells for the solid phase is fixed to Nw =9. The kinematic viscosity of the fluid is − fixed to ν =5⇥ 10 5 and the Ekman friction parameter to α =0.005.

estimating mass fluxes between adjacent cells. We make use of a positivity-preserving flux-limiter algorithm in order to avoid diverging gradients in the distribution (see, e.g., LeVeque 2002). One of the key points of this numerical method is to provide a priori a correct guess for the maximum velocity of the particles Wmax. To avoid arbitrary cautious choices that might lead to non-optimal resolutions, we adopt an adaptive re-meshing of the velocity grid, by keeping fixed Nw and imposing Wmax to be at least twice the root-mean squared value of the particle velocity computed from the distribution f. The time step ∆t is then adapted to maintain the constraint ∆x = ∆w ∆t. Here and in the following, the system is discretized on a square domain of size L = 2π with periodic boundary conditions. The particles are integrated with a resolution 2 2 2 2 Nx ⇥ Nv = 1024 ⇥ 9 and the fluid velocity discrete Fourier transform is computed with 5122 wavenumbers (corresponding to an effective resolution of 10242 collocation points). The flow is sustained by a large-scale Gaussian random forcing, acting on modes k with moduli satisfying 1 < |k| < 4. The simulations are initialized with the fluid flow at rest and uniformly distributed particles with zero velocity. Once the flow and the particles have reached a statistically stationary regime, the statistics are performed over 300 large-eddy turnover times T = L/urms. The parameters of the simulations are reported in Table 1. In Figure 2 we compare typical snapshots (taken at random times in the stationary regime) of the fluid vorticity ω(x,t) (upper panels), along with the particle density field np(x,t) integrated over velocities (lower panels), with and without back-reaction 8 F. Laenen, S. Musacchio, J. Bec

Figure 2. Snapshots of the fluid vorticity (upper panels) and of the particle density (lower panel) illustrating the effect of the coupling for St0 ' 17. The left panels correspond to the case with no coupling between the two phases (φm = 0). The right panels are for the case with a moderately strong coupling (φm =0.6). The color scales are the same for the left and right figures.

from the particles and in both case for St0 ' 17. In the case without coupling (left panels), the fluid flow is dominated by large eddies and particles are ejected from them to concentrate in the high-strain filamentary regions outside. This is a clear illustration of preferential concentration by centrifugal effects. Note that for such a high value of the Stokes number, the particle density is nearly uniform. Preferential concentration is indeed weak as the particles tend to follow ballistic trajectories, ignoring the vortices. As soon as the coupling is turned on (right panels), one clearly detect a significant damping of the large scales, still keeping larger densities outside the eddies. This qualitative observation can be quantitatively confirmed by measuring the energy-containing scale defined as

LE(φm)=( k E(k)/k)/( k E(k)) where E(k) is the energy power spectrum of the fluid velocityPu. This lengthscale,P which is not shown here, is a decreasing function of the mass load φm. As can been seen from the lower panels of Fig. 2, the particle density pictures an enhancement of the fluctuations, indicating an increase of clustering due to the coupling. 9

St =1 1 (a) 0 1 (b) St0 =4.5 St0 = 17 0.8 0.8 I 0 η / C

Z/Z 0.6 0.6 η , , 0 I ε / /E C E 0.4 ε 0.4

0.2 0.2 St0 =1 St0 =4.5 St0 = 17 0 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 φm φm

Figure 3. (a) Total energy E (dashed lines, open symbols) and enstrophy Z (solid lines, filled symbols) normalized by their respective values E0 and Z0 obtained with no coupling, and represented as a function of the mass loading φm for various values of the Stokes number as labeled. (b) Average rates of energy exchange εC (dashed lines, open symbols) and of enstrophy exchange ηc (solid lines, filled symbols) between the fluid and the particle phases. These quantities, normalized to their respective injection rates εI and ηI , are represented as a function of the mass loading φm for the different values of the Stokes number.

0.7 1 St St (a) 0 =1 0 =1 St =4.5 0.9 (b) St =4.5 0.6 0 0 St = 17 St = 17 0 0.8 0

0.5 0.7 I I η η 0.6 / / 0.4 ν α η η , , 0.5 I I ε ε / 0.3 / ν α 0.4 ε ε 0.2 0.3 0.2 0.1 0.1

0 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 φm φm

Figure 4. (a) Average dissipation rates of energy εν (dashed lines, open symbols) and of enstrophy ην (solid lines, filled symbols) due to molecular viscosity, normalized to the injection rates εI and ηI , and represented as a function of the mass loading φm for the different values of the Stokes number. (b) Same for the dissipation rates εα and of ηα due to the linear Ekman friction term.

4. Modification of energy and enstrophy budgets We report in this section measurements on how the energy and enstrophy of the fluid are affected by the coupling with the particle phase. The feedback of the particles has strong consequences on the global kinetic energy E = (1/2)h|u|2i and the enstrophy Z = (1/2)hω2i, where the brackets h·i denotes averages over space and time. In particular, as seen in Fig. 3(a), we find that both quantities decrease at increasing the mass loading ratio φm. The reduction of the kinetic energy of the fluid phase can be understood by considering the global energy balance, which in the statistically steady state reads

εI = εα + εν + εC . (4.1) 2 Here, εI = hFu · ui denotes the average energy input, εα = α h|u| i is the energy 10 F. Laenen, S. Musacchio, J. Bec 2 dissipation rate due to friction, εν = ν h|ru| i is the energy dissipation rate due to 2 viscosity, and εC = hFp!f · ui = −φm hnp(|u| − u · vp)i/τp is the exchange rate of kinetic energy from the fluid to the particles. One can show analytically (see Appendix for details) that the coupling between the fluid and particle acts on average as a dissipative term for the kinetic energy of the flow, that is hFp!f · ui > 0. This can be understood intuitively from the simple consideration that the particles move because the flow gives them a fraction of its kinetic energy. This is confirmed by the numerical simulations, which shows a significant transfer of energy from the fluid to the particle as soon as φm > 0. This effect increases with the mass loading φm, as can be seen in Fig. 3(b). Already when the mass ratio is φm =0.1, we find that more than 50% of the kinetic energy injected in the system is transferred to the particles. The viscous dissipation rate εν , being proportional to the enstrophy Z, decreases with φm as confirmed in Fig. 4(a). As usual in the two-dimensional direct enstrophy cascade, its contribution to the energy balance is sub-dominant and this effect is unvarying when the coupling increases: We always observe εν /εI < 2% with almost no dependence on φm or St0. The dissipation due to friction εα is actually dominating the energy budget. Because it is proportional to the kinetic energy, we expect it to decrease with φm, as confirmed in Fig. 4(b). In analogy to the quantities introduced in the energy balance (4.1), we define the enstrophy input rate ηI = hFω ωi, the enstrophy dissipation rate due to viscosity, ην = 2 2 ν h|rω| i, the enstrophy dissipation rate due to friction ηα = α hω i, and the exchange rate of enstrophy with the particle phase ηC = hr⇥ Fp!uωi. Their values obtained from the various simulations are reported in Tab. 1. As for energy, these quantities enter in the enstrophy budget, which reads in the statistically steady

ηI = ηα + ην + ηC (4.2)

The dissipation rate of enstrophy due to friction ηα is proportional to the enstrophy itself, and therefore displays the same slowly decreasing behavior as a function of φm, as shown in Fig. 4(b). The contributions ην and ηC stemming from molecular viscosity and coupling with the particle phase reveal a much richer phenomenology. We see from Fig. 4(b) that the viscous dissipation decreases with φm at large values St0, while it displays a non- monotonic behavior in the case St0 ' 1. Similarly, the transfer from the enstrophy of the fluid to the particles increases with φm only at large St0, while the case St0 ' 1 displays a non-monotonic behavior, as visible in Fig. 3(b). As we will discuss in the following, the increase at large φm of both the enstrophy viscous dissipation comes from a strong coupling with the particle phase that enhances the enstrophy transfers to the small scales. In our simulations the flow is sustained by a large-scale forcing, which gives rise to an enstrophy cascade toward viscous scales. We investigate here in more details how the coupling with the particle phase influences this process by measuring the scale-by-scale enstrophy budget. The equation for the enstrophy spectrum Z(k) reads:

2 ∂tZ(k)=−T (k) − 2α Z(k)+ΦZ (k) − 2ν k Z(k)+C(k), (4.3) where ΦZ (k)=hF[Fω ω]i is the spectral injection of enstrophy by the forcing, F[·] denoting the Fourier transform. This contribution is concentrated at scales k<4. The term C(k)=hF[r⇥Fp!u ω]i is the spectral contribution due to the coupling with particles and T (k)=hF[ω (u · rxω)]i is the spectral enstrophy transfer due to the non-linear advection term. The averages are taken over time and the wave-number shell of modulus |k| = k. In the statistically stationary regime, ∂tZ(k) = 0 and the various terms on the right-hand side compensate each other. In the inertial range and 11 -3 ×10-3 ×10 ×10-3 5 5 5 T (k) T (k) (b) (c) T (k) (a) C(k) C(k) C(k) 2ν k2 Z(k) 2ν k2 Z(k) 2ν k2 Z(k)

0 0 0

-5 -5 -5 101 102 101 102 101 102 Wavenumber k Wavenumber k Wavenumber k

Figure 5. Spectral contributions of non-linear transfer T (k), coupling with the particle phase C(k), and viscous dissipation 2νk2Z(k) entering in the enstrophy balance (4.3) for (a) no coupling: φm = 0, (b) coupling with a large Stokes number: φm =0.5 with St0 ' 17, and (c) coupling with a moderate Stokes number: φm =0.5 with St0 ' 1. in absence of coupling, the non-linear transfer is balanced by viscous dissipation, i.e. T (k) '−2ν k2 Z(k). The spectral behaviors of the viscous term, the transfer term and the coupling term are shown in Fig. 5. In the absence of coupling (φm = 0, left-most panel), we observe a balance in the inertial and dissipative ranges between the non-linear transfer and the viscous dissipation. In the case of strong coupling and a large value of the Stokes number (φm =0.5, St0 ' 17, middle panel), we observe at low wavenumbers an important dissipative (i.e. negative) contribution from the term due to coupling, which is balanced by a symmetric positive enhancement in the enstrophy transfer term. It seems in that case that all the effects of the particles back-reaction are present over all scales, even if they monotonically decrease with k. For the case of a smaller Stokes number (St ' 1, right-most panel), the spectral behavior is much more complex. In this case, the effects of coupling is still dissipative at very low wavenumbers, but becomes positive in an intermediate range of wavenumbers. This means that the interaction between the two phases is responsible for injecting enstrophy at intermediate scales. Consequently, the transfer term also changes sign. In the limit of small Stokes numbers it is possible to show that this phenomenon is connected to the occurrence of small-scales instabilities. The numerical method used in this work is however not satisfactorily addressing the asymptotics of weak particle inertia, so that undertaking this issue requires appropriate developments. This is thus the subject of a separate work that will be published elsewhere. As observed in Fig. 5(b), the back-reaction of large-Stokes-number particles strongly modifies the enstrophy scale-by-scale budget. The coupling acts over all scales and breaks up the balance between non-linear transfer and viscous dissipation. This effect causes significant changes in the enstrophy power spectrum. Figure 6(a) shows the various enstrophy spectra obtained when varying the mass load φm while keeping the Stokes number constant at a fixed large value St0 ' 17. For a better visibility, the spectra are compensated with the slope measured in the uncoupled case, namely Z(k) / k−1.4. The deviation to the k−1 prediction of Kraichnan (1967) is due to the presence of linear damping. It is indeed known that in two-dimensional turbulence with Ekman friction, the competition between the exponential separation of Lagrangian trajectories and the exponential damping of fluctuations is responsible for a strong intermittency, and in particular drastically affects the spectral scaling of enstrophy (Boffetta et al. 2005). The 12 F. Laenen, S. Musacchio, J. Bec ) ) k k ( (a) ( (b) Z 0 Z 0 4 8

. 10 . 10 1 1 k k

10-1 10-1

10-2 10-2

φm =0 φm =0.1 . 10-3 φm =02 10-3 φm =0.3 φm =0.4 φm =0.4 with α =0.005 φm =0.5 φm = 0 with α =0.04 -4 -4 Compensated enstrophy10 spectrum Compensated enstrophy10 spectrum 100 101 102 100 101 102 Wavenumber k Wavenumber k

Figure 6. (a) Fluid enstrophy spectra for St 0 ⇠ 17 fixed and various mass loads; the spectra are compensated with the power-law k1.4 corresponding to the case with no coupling.(b) Comparison of the enstrophy spectra obtained in the coupled case with St = 17 and φm =0.4 and in the uncoupled case with the effective friction given by (4.4). The spectra have been compensated by k1.8 for better visualization. power-law scaling of Z(k) is steeper than the k−1 prediction, i.e. Z(k) / k−1−δ, and the deviation δ > 0 increases linearly with the Ekman friction coefficient α (Verma 2012). We see from Fig. 6(a) that when increasing the mass load φm and thus the strength of coupling with the particles, an additional steepening of the spectrum power-law exponent occurs. This effect can be explained by an heuristic argument. At very large Stokes, the particles are almost uniformly distributed, np(x,t) ' 1/V, and their velocity is on average much smaller than that of the fluid |vp| ⌧ |u|. This suggests that the coupling force (2.8) entering in Navier–Stokes equation (2.4) can be approximated as

φm φm Fp!f = − np [u − vp] ⇡− u. (4.4) τp τp V

Coupling thus acts to leading order as an effective linear friction with coefficient αeff = φm/(τp V). From a phenomenological viewpoint, this amounts to say that the particles are seen by the fluid almost like fixed obstacles which are increasing the friction drag. In order to check the above arguments predicting that the modification of the enstrophy spectrum by particles can be mimicked by an increase of the friction coefficient α, we compare in Fig. 6(b) the results form a simulation with St = 17 and φm =0.4 with those obtained in the absence of coupling but with the effective friction given by (4.4). The collapse of the two spectra in the scaling range and viscous range is remarkable. This indicates that the effect of the particles on the vorticity filament in the straining regions can be effectively modelled by an increased friction. However we observe some discrepancies at large scales. The increase of the effective friction indeed causes a strong depletion of the large-scale vortices. Conversely, the particles are ejected from the large- scale vortices. Therefore they causes weaker modifications of the flow at those scales.

5. Amplification of intermittency In previous section, we have focused on the second-order statistics of the velocity and vorticity fields. Here we investigate the effect of two-way coupling on higher-order statistics. In particular, we study the statistics of the vorticity increments δrω = ω(x + r) − ω(x) and of their moments, which define the vorticity structure functions Sp(r)= p h|δrω| i, where the angular brackets comprise averages over space, time and, by isotropy, 13

102 102 (a) (b) p p rms rms ω ω / / ) ) r r ( ( p p S 100 S 100

10-2 10-2

p =1 p =1 p =2 p =2 p p -4 =3 -4 =3 10 p =4 10 p =4 p =5 p =5 Vorticity structure functions Vorticity structure functions

10-3 10-2 10-1 10-3 10-2 10-1 r/L r/L

Figure 7. Vorticity structure functions of order p =1...5 (as labeled), represented as a function of the separation r in the cases (a) without coupling (φm = 0) and (b) with φm =0.5 and St0 ' 17.

2 2 (a) (b) 1.8 1.8

1.6 1.6 p ζ 1.4 1.4

1.2 1.2 2 ζ

1 / 1 p ζ 0.8 0.8

0.6 0.6 Scaling exponents φ = 0 with α =0.005 0.4 0.4 m φm =0 φm =0.2 with α =0.005 0.2 φm =0.2 0.2 φm =0.5 with α =0.005 φm =0.5 φm = 0 with α =0.04 0 0 012345012345 Order p Order p

Figure 8. (a) Scaling exponents ζp of the p-th order vorticity structure function for various values of the mass loading φm and for St0 ' 17, obtained by averaging the logarithmic derivative d log Sp(r)/d log r over the range 0.015

15 100 φm =0 (b) φm =0 φm =0.2 φm =0.2 . . φm =05 -1 φm =05 Gaussian 10 2 )] r (

2 10 S [ 10-2 / ) r ( 4 S 10-3 5

Flatness -4

Probability density10 function

(a) 0 10-5 10-3 10-2 10-1 -10 -5 0 5 10 ∂ ω ∂ ω 2 1/2 r/L r /⟨( r ) ⟩

2 4 2 2 Figure 9. (a) Flatness F(r)=S4(r)/[S2(r)] = hδrω i/hδrω i of the distribution of vorticity increments as a function of the separation r and for St0 ' 17 and various values of the mass load. (b) Probability density function of the vortcity gradients normalized to unit variance represented here for St0 ' 17 and various values of the mass load.

in agreement with what was observed in previous section for second-order statistics. However, the dependence on φm remarkably disappears once the exponents are rescaled by the value of ζ2, a representation which is somewhat equivalent to the extended self- similarity approach. As can indeed be seen in Fig. 8(b), the various measurements of ζp/ζ2 collapse, within statistical uncertainty, onto a unique universal curve. The measurement obtained from a different value of the linear damping coefficient α are put on the top of the measurements of Fig. 8(b). They also follow the same universal law. This observation suggests, on the one hand, that all intermittency effects due to Ekman friction on the inertial-range scaling exponents of vorticity are entailed in second-order statistics, i.e. one has ζp(α)/ζ2(α)=f(p) where f(p) is a universal function independent of the coefficient α and, on the other hand, the coupling with a particle phase associated to a large Stokes number is reproduced at all orders by the effective linear friction expressed in (4.4). This specific behavior in inertial-range multiscaling properties is not detectable from usual measurements of intermittency. For instance, it is frequent to make use of the 2 flatness of the distribution of increments defined as F(r)=S4(r)/[S2(r)] , whose discrepancy to the value F(r) = 3 quantifies the deviations from a normal distribution. The changes in behavior of F(r) at varying the coupling with the particle phase are represented in Fig. 9(a). Above results suggest that, in the inertial range, F(r) ⇠ rγ with γ =(f(4) − 1) ζ2, with all effects of coupling entailed in the variations of ζ2. This completely explains the increase of F(r) observed when fixing r in the inertial range and increasing φm. However, Figure 9(a) suggests that the effects of the coupling are not limited to the scaling range. At large scales, we observe that the flatness of vorticity increments decreases with φm. This behavior is connected to the suppression of the large scales eddies, already observed qualitatively in Fig. 2. Conversely, at dissipative scales, we observe that F(r) attains in all cases a plateau with stronger departures from a Gaussian distribution when the coupling with particles and thus φm increases. This is confirmed in the limit r ! 0 by measuring the probability density functions (pdf) of the vorticity gradient. The results are shown in Fig. 9(b). When φm increases, the pdfs develop broader tails even if the vorticity gradient is normalized with its standard deviation. This effect suggest that, at difference with the inertial range, the effect of particles on small-scale statistics is not fully determined by second-order quantities. 15

1.1

St0 =1 St0 =4.5 1 St0 = 17

0.9 0 St

/ 0.8 St

0.7

0.6

0.5 0 0.1 0.2 0.3 0.4 0.5 φm

Figure 10. Stokes number St = τp ωrms represented here normalized by its reference value St0 obtained from the uncoupled regime and as a function of the particle mass loading φm.

6. Enhancement of particle clustering To go beyond the modulation of turbulence by particles, we address in this section the question of how the particle spatial distribution is itself affected by two-way coupling. A first prediction can be made by recalling that we observed a decrease of the total enstrophy when the particle mass loading is increased. As a result, the time scale of the flow increases and the effective Stokes number experienced by the particles becomes smaller than St0. This is shown in Fig. 10 which represents the evolution of the actual Stokes number St as a function of the intensity of coupling. We observe that St decreases monotonically with φm for all St0 considered, the strongest reduction being of the order of 40%. As a straightforward consequence and since St0 & 1, this effect goes together with an increase of particle concentration. A reduction of St indeed brings the distribution of the particles closer to the maximum of clustering, which is known to occur at St ⇠ 1. In order to test this prediction, we report in Fig. 11(a) the measurement of the probability density function of the particle spatial density np for different intensities of the back-reaction and for the largest available Stokes number, i.e. St ' 17. In the absence of coupling, that is when φm = 0, the distribution is very narrow around its 2 average hnpi = n0 =1/L , which, by conservation of mass and spatial homogeneity of the statistics, is independent of both φm and St. The observed narrow distribution indicates that the spatial repartition of the mass is nearly uniform, as it is expected for particles with large Stokes numbers. This is in agreement with the observed instantaneous density field displayed in Fig. 2 (lower left panel). When the coupling is turned on, that is when φm > 0, the distribution significantly broadens. Regions with very few or with high particle-number become more probable, so that clustering is enhanced. We note that this effect is much stronger than what would be expected by the sole reduction of the Stokes number. One notice that the distribution of particle density develops a power-law tail at small values np ⌧ n0. The exponent associated to this decreases when the intensity of coupling increases. It reveals an increasing contribution of voids in the statistics of the particle spatial density. 2 The intensity of clustering can be quantified by the variance h(np − n0) i of density fluctuations. This quantity is represented as a function of φm in Fig. 11(b) for the various reference Stokes numbers St0 considered in this work. The error bars are computed as follows: for every time series of the second order moment of the field np(x,t), we compute 2 a sliding mean m(t)=hhnp(x,t)it2[T0,T ]ix with an increasing windowing size τ = T −T0. T0 is the time at which the distribution reaches a statistically stationary state. A time 16 F. Laenen, S. Musacchio, J. Bec

2

φm =0 (b) St0 =1 (a) 1.8 φm =0.2 St0 =4.5 0 φ =0.5 St = 17 10 m 1.6 0

1.4

1 1.2 −

-1 2 0 10 1 /n ⟩ 2 p

n 0.8 ⟨ 0.6 -2 10 0.4 Probability density function 0.2

0 10-1 100 0 0.1 0.2 0.3 0.4 0.5 φ np/n0 m

Figure 11. (a) Probability density function of particle density ρp(x) represented for different 2 φm and St0 = 17. Note that hρpi =1/L is independent of φm. (b) Variance of the density fluctuations as a function of φm for St0 ⇠ 17. t = T ⇤ is chosen at which the average m(t) seems to be stationary. The error is estimated ∗ as maxt2[T ,Tmax] |m(t) − m(Tmax)|. For the highest Stokes number (bottom line with triangular symbols) and as pointed out earlier, those fluctuations indeed increase as a function of the mass loading φm. This is also true for the intermediate Stokes number. However, at the lowest value, the behavior changes. We indeed observe for St0 ' 1a decrease of clustering when the coupling intensity increases. This effect can be partially explained by the reduction of the Stokes number which brings the particles away from the maximum of clustering. Another effect which can contribute to the same phenomenon may be due to small scale instabilities, which occur at small Stokes and that were already mentioned in §4. These instabilities are responsible for disrupting the large-scale vortices of the flow from which the particles are ejected. This mechanism thus leads to an increase of particle mixing.

7. Summary and conclusion We have investigated in this work the modulation of turbulence by heavy, point- like particles in the two-dimensional direct enstrophy cascade. We made use of a field formulation in position-velocity phase space for the particle dynamics, following the recent development in Laenen et al. (2016) of a numerical method whose validity has been assessed at large Stokes numbers. The fluid flow being characterized by a unique timescale, particles dynamics in the direct enstrophy cascade is fully characterized by a unique Stokes number. At large values of the Stokes number, we found that the effects of two-way coupling are fully reproduced by an effective Ekman damping associated to an extra linear friction term in Navier–Stokes equation. This purely dissipative effect disappear at Stokes numbers of the order of unity or smaller. The particle phase still acts as an enstrophy pump at large scales but is able to transfer and re-inject it at smaller scales. As a result of this, the non-linear transfer over scales is reversed leading to a deep qualitative change of the direct cascade phenomenology. We also observed in this work a marked effect of coupling on the intermittency of the fluid flow. In the scaling range, this effect is actually a straightforward consequence of the modulation of second-order statistics. We have indeed seen that the scaling exponents of the vorticity structure functions collapse to a universal curve, once they are rescaled by the second-order exponent, and independently of either the particle mass loading or 17 the coefficient of Ekman damping. At small scales, the coupling with a particle phase is responsible for a flattening of the probability density function of vorticity gradients. Finally, it was shown that the two-way coupling also impacts clustering. The distribution of the particle spatial density field develops larger tails as the coupling intensity increases. This effect might be partly attributed to the reduction of particles relative inertia due to the attenuation of enstrophy fluctuations in the flow. A clear extension of this work could consist in repeating such a study in the case of the inverse energy cascade. In that case, characteristic timescales τr depend on the observation scale r and monotonically increases with it as r2/3. Hence, the strength of particle inertia is measured by a scale-dependent, local Stokes number St(r)=τp/τr. When the scale associated to St(r) = 1 falls inside the inertial range, the small scales are impacted by particles with St < 1, while large scales feel particles with St > 1. This would lead to an intricate situation where both phenomenologies are present. Such settings might clearly shed lights on what could happen in a fully developed, three- dimensional turbulent flow.

This work benefited from useful discussions with Giorgio Krstulovic who is warmly acknowledged. This research has received funding from the French Agence Nationale de la Recherche (Programme Blanc ANR- 12-BS09-011-04). Simulations were performed using HPC resources from the Mesocenter SIGAMM hosted by the Observatoire de la Cˆote dAzur.

Appendix A. Energy conservation and dissipative effect of particles

We derive here a prediction for the sign of εC which is the contribution in the fluid kinetic energy budget coming from the interaction with the particles — see equation (4.1). 2 We denote by Ef (x,t)=|u(x,t)| /2 the spatial fluid kinetic energy field, Ep(x,t)= 2 2 f(x, w,t)|w| d w/2 that of the particle phase and Ep = hEp(x,t)i the average energy ofR the particles. Here and in the following the average is over time and the spatial domain. We first write down the definition of εC : 1 ε = − hn v · ui−hn |u|2i (A 1) C τ p p p p ⇥ ⇤ Using the Liouville equation (2.2), the conservation of particle kinetic energy reads:

dEp 2 2 1 = h ∂tf |w| d wi = [hnpvp · ui−2hEpi] (A 2) dt Z τp Furthermore, using Holder inequality, it can be shown that 2 f|w|2 d2w ⇥ f d2w > fw d2w , so that 2E > n |v |2 (A 3) Z Z �Z � p p p � � � � Since dEp/dt > 0 (particles are initially� at rest), equation� (A 2) implies 2 hnpvp · ui > 2hEpi > hnp|vp| i (A 4) 2 We further notice that hnpvp · ui cannot be simultaneously greater than hnp|u| i and 2 2 hnp|vp| i because of the Cauchy-Schwartz inequality. Hence hnp|u| i > hnpvp · ui and εC > 0, meaning that two-way coupling may only withdraw kinetic energy from the fluid.

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J´er´emie Bec,1 Fran¸cois Laenen,1 and Stefano Musacchio2 1Universit´eCˆote d’Azur, CNRS, OCA, Laboratoire J.-L. Lagrange, Nice, France 2Universit´eCˆote d’Azur, CNRS, Laboratoire J.-A. Dieudonn´e, Nice, France The feedback foces exerted by particles suspended in a turbulent flow is shown to lead to a new − scaling law for velocity fluctuations associated to a power-spectra / k 2. The mechanism at play relies on a direct transfer of kinetic energy to small scales through Kelvin–Helmholtz instabilities occurring in regions of high particle density contrast. This finding is confirmed by two-dimensional direct numerical simulations.

It is common to face environmental, industrial or as- density ρp and a particle velocity field vp satisfying trophysical situations where impurities such as dust, ∂ ρ + r · (ρ v )=0 (2) droplets, sediments, and other kinds of colloids are trans- t p p p 1 ported by a turbulent fluid. When the suspended parti- ∂tvp +(vp · r) vp = − (vp − u) , (3) cles have finite sizes and masses, they detach from the τp 2 flow by inertia and form uneven distributions where in- where τp =2ρs a /(9ρf ν) is the particles response time, tricate interactions and collisions take place. The physi- a being their radius and ρs the mass density of the mate- cal processes at play are rather well established, leading rial constituting the particles. The hydrodynamical sys- to quantitative predictions on the rates at which cloud tem (2)-(3) has proven to be a valid approximation for droplets coalesce [1], dust accrete to form planets [2], or relatively small Stokes numbers St = τp/τf [9], that is heavy sediments settle in a turbulent environment [3, 4]. when the particle response time is smaller than the small- Still, basic and important questions remain largely est active timescale τf of the fluid flow. In this limit, open as to the backward influence of particles on the fold caustics appear with an exponentially small proba- carrier flow structure and geometry. Some situations bility [10, 11], preventing the development of multivalued involve particle mass loadings so large that the fluid branches in the particle velocity profile and thus ensuring turbulent microscales are altered and, in turn, several the validity of a hydrodynamical description. macroscopic processes are drastically impacted. These The force exerted by the particles on the fluid reads include spray combustion in engines [5], aerosol salta- 1 ρp tion in dust storms [6], biomixing by microorganisms in f ! = (v − u) . (4) p f τ ρ p the oceans [7], and formation of planetesimals by stream- p f ing instabilities in circumstellar disks [8]. Currently such It is proportional to the mass density of the dispersed systems are unsatisfactorily handled by empirical ap- phase and thus combines the heaviness of the particles proaches or specific treatments. A better modelling re- with their number density. The strength of feedback is quires identifying and understanding the universal phys- measured by the comprehensive non-dimensional param- ical mechanisms at play in turbulence modulation by dis- eter Φ = hρpi/ρf . It involves the particle density spatial persed particles. In this spirit, we focus here on the alter- average hρpi = Np mp/V, where Np is the total number ation of small scales by tiny heavy spherical particles. We of particles, mp their individual mass, and V the volume show that the fluid velocity is unstable in regions with a of the domain. All these quantities being conserved by high particle density contrast, leading to energy transfers the dynamics, so is the coupling parameter Φ. shortcutting the classical turbulent cascade. This effect We first draw some straightforward comments pertain- leads to a novel scaling regime of the turbulent velocity ing to the limit of small Stokes numbers. There, particles field associated to a power-law spectrum / k−2. almost follow the flow with a tiny compressible correc- The fluid velocity field u solves the incompressible tion [12], namely vp ⇡ u−τp a, where a = ∂tu+(u·r)u Navier–Stokes equations: r · u = 0 with denotes the fluid flow acceleration field. The feedback force exerted on the fluid is hence, to leading order, 1 2 ∂tu +(u · r) u = − rp + ν r u + fext + fp!f . (1) ρf 1 fp!f (x,t) ⇡− ρp(x,t) a(x,t) (5) ρf ρf is here the fluid mass density and ν its kinematic viscosity. A homogeneous isotropic turbulence is main- The effect of particles can thus be seen as an added mass, tained in a statistical steady state by an external forcing which does not depend upon their response time and is fext. The fluid flow is perturbed by a monodisperse popu- responsible for an increase of the fluid inertia. The fluid lation of small solid particles whose effects are entailed in is accelerated as if it has an added density equal to that of the force fp!f . These particles are assumed sufficiently the particles. Such considerations predict that the pres- small, dilute and heavy for approximating their distri- ence of particles decreases the effective kinematic viscos- bution and dynamics in terms of fields, namely a mass ity of the fluid and thus increases its level of turbulence. 2

Φ =0 Φ =0.4 This vision is however too naive as it overlooks the spatial 30 fluctuations of the particle density. It is indeed known that an even infinitesimal inertia of the particles creates 20 extremely violent gradients of their density through the 10 mechanism of preferential concentration. As we will now see, these variations are responsible for instabilities that 0 shortcut the turbulent energy cascade by directly trans- -10 ferring kinetic energy to the smallest turbulent scales. A key attribute of turbulence is the vigorous local spin- -20 ning of the fluid flow, weighed by the vorticity ω = r⇥u. -30 The effect of particles on the vorticity dynamics is en- 1.8 tailed in the curl of the feedback force (4), reading 1.6 1 1.4 r⇥fp!f = [ρp (ωp − ω)+rρp ⇥ (vp − u)] , (6) 1.2 ρf τp 1 where ωp = r⇥vp is the vorticity of the dispersed phase. 0.8 The action of particles is thus twofold. The first term ac- 0.6 counts for a friction of the fluid vorticity with that of 0.4 the particles, which amounts at small Stokes numbers 0.2 to the above-mentioned added-mass effect. The second 0 term gives a source of vorticity proportional to the gra- dients of particle density. The combined effects of prefer- FIG. 1. (color online) Upper frames: Snapshots of the ential concentration and turbulent mixing is responsible (scalar) fluid vorticity ω = r⇥u in two dimensions for both the passive case with no feedback (Φ = hρpi/ρf = 0, left) and for very sharp spatial variations of ρp. Centrifugal forces indeed eject heavy particles from coherent vortical struc- when the particles exert a back reaction on the flow (Φ =0.4, right). Lower frames: normalized particle density ρp/hρpi at tures [13] and Lagrangian transport stretches particles the same instants of time. The two cases (without and with patches in stirring regions [14]. This leads to the devel- coupling) correspond to different realizations of the external opment of substantial fluctuations of rρp, as illustrated forcing; the x and y axis were however shifted in order to in two dimensions on the left panels of Fig. 1. This mech- locate large-scale structures at approximately the same posi- anism creates regions with very strong shear in the fluid tion. flow, which, in turn, develop small-scale vortical struc- tures through Kelvin–Helmholtz instability. It is indeed well known that flows presenting a quasi-discontinuity of term on the right-hand side) and viscous dissipation (sec- velocity are linearly unstable and develop wavy vortical ond term), leading for isotropic flows to the celebrated streaks at the interface of the two motions (see, e.g., [15]). Kolmogorov 4/5 law. In the presence of coupling with Such phenomenological arguments thus suggest that the particles, this equilibrium is broken by the feedback force. feedback of particles lead to the formation of small-scale In the asymptotics St ⌧ 1 of low inertia, this force is ap- eddies, as can be seen in the right panels of Fig. 1. Par- proximated by (5), so that its contribution to (7) reads ticles thus actively participate in the transfer of kinetic energy toward the smallest turbulent scales. 1 1 This effect and the resulting modification of the fluid hfp!f · u¯i⇡ hρp a · ui− hρp a · δrui . (8) ρf ρf flow scaling properties can be quantified by examin- ing the scale-by-scale kinetic energy budget given by The first term on the right-hand side involves the corre- K´arm´an–Howarth–Monin relation (see, e.g., [16]). De- lation between the particle density field and the instan- noting the velocity increment over a separation r by taneous power acting on fluid elements. To leading order 0 0 δru = u − u with u = u(x + r,t) and u = u(x,t), when St ! 0, we have hρp a · ui⇡hρpiha · ui = 0. Non- one can easily check that statistically homogeneous solu- vanishing corrections at small but finite Stokes numbers tions to the Navier–Stokes equation (1) satisfy might arise from a combined effect of the small compress- ibility of the particle velocity together with the biased 1 0 1 2 2 0 ∂t hu · u i = rr · |δru| δru + ν r hu · u i sampling due to preferential concentration, as already 2 4 r seen for the radial distribution function [17, 18]. How- + u · Df¯ + hf !E · u¯i , (7) ext p f ever such correlations are in the best case of the order ⌦ ↵ where the overbar denotes the average over the two points of St2. The second term on the right-hand side of (8) located at x ± r, that is f¯ =[f(x + r,t)+f(x − r,t)]/2. does not vanish in the limit St ! 0 and thus gives the In classical stationary turbulence, the above relation sug- dominant contribution. gests a balance between the non-linear transfer term (first Such arguments lead to predict that the scale-by-scale 3 energy balance (7) reduces in the inertial range to ) = 0 ) 100 ) = 0:1 k 1 2 1 ( ) = 0:2

r · |δ u| δ u ' hρ a · δ ui . (9) E r r r p r ) 4 D E ρf = 0:4 10-2 Now, assuming that the fluid velocity field obeys some slope = -2 h scaling property δru ⇠ r , one deduces from the above balance that 3h−1=h, and thus h =1/2. Such a scaling 10-4 behavior is associated to an angle-averaged kinetic energy slope = -3.3 power spectrum E(k) / k−2. In order to test such prediction, we perform two- 10-6 dimensional simulations of the fluid-particle system de- fined by (1), (2), (3), and (4) in a periodic domain. We 2 -8 make use of a Fourier-spectral solver with 1024 collo- Fluid kinetic energy10 spectrum cation points for estimating spatial derivatives and of a second-order Runge-Kutta scheme for time marching. 100 101 102 We focus on the direct enstrophy cascade, so that the Wavenumber k external forcing fext is the sum of an Ekman friction with timescale 1/α and of a random Gaussian field η FIG. 2. (color online) Angle-averaged kinetic energy power white noise in time and concentrated at wavenumbers spectra of the fluid velocity represented for various values of |k|  2. We make use of hyper-viscosity and hyper- the coupling parameter Φ, as labelled. diffusivity (fourth power of the Laplacian) in order to maximize the extent of the inertial range and prevent Eqs. (2) and (3) from blowing up. The particle response pacts only the largest scales of the flow and the smaller time is fixed in such a way that St = τ hω2i1/2 ⇡ 10−2 p scales experience an increase in their energy content. The in the uncoupled case and various values of the coupling inertial-range is characterized by a shallower power spec- parameter Φ = 0, 0.1, 0.2, and 0.4 are simulated. trum with an exponent close to −2, as expected from Figure 1 shows snapshots of the fluid vorticity field to- above arguments. Dissipative scales are shifted toward gether with the particle density field, without and with larger wavenumbers, as a consequence of the added-mass coupling between the two phases. In the absence of feed- effect induced by particles which decreases the effective back from the particles (left panels), the flow develops the kinematic viscosity of the fluid loaded by particles. traditional picture of two-dimensional direct cascade con- sisting of large-scale vortices separated by a bath of fil- #10-4 amentary structures where enstrophy is dissipated. The 4 particles density field is characterized by large voids in Coupling Transfer 3 the vortical structures separated by a filamentary distri- Dissipation bution that is symptomatic of turbulent mixing. These Friction qualitative pictures are strongly altered when the particle 2 feedback is turned on. In the presence of coupling (right panels), the fluid flow still shows large-scale structures 1 but which are this time surrounded by a bath of small- scale vortices. These eddies form wavy structures along 0 the lines associated to quasi-discontinuities of the parti- cle density field. This is a clear signature that Kelvin– -1 Helmholtz instability is at play. Scale-by-scale energy budget Figure 2 shows the angle-averaged power spectra of -2 the fluid kinetic energy obtained when varying the cou- pling parameter. In the case of no feedback (Φ = 0), the -3 specific choices of the Ekman coefficient α and of the en- 101 102 ergy injection amplitude yield a kinetic energy spectrum Wavenumber k E(k) / k−δ with δ ⇡ 3.3. For any non-vanishing value of the coupling parameter Φ, one observes remarkable FIG. 3. (color online) Angle-averaged Fourier amplitudes of the various terms contributing to the kinetic energy bud- changes in the spectral behavior of the fluid velocity. The ff get (7) shown here for Φ =0.4. Coupling stands from the first e ect is a clear decrease of the total kinetic energy. contribution of the forces exerted by the particles on the fluid, Similarly to what is obtained in the asymptotic of large transfer for the nonlinear advection terms, dissipation for vis- Stokes numbers [19], this is due to a net dissipative effect cous forces and friction for Ekman damping. of the coupling with the particle phase. However this im- 4

Further insight is given by measuring the amplitude a more systematic manner: They might indeed strongly of the various terms entering in the energy budget (7). modify at both qualitative and quantitative levels the Figure 3 shows the angle-averaged amplitude of their rate at which particles interact together. Fourier transforms with respect to the separation r. One We acknowledge useful discussions with G. Krstulovic. observes that the non-linear transfer term gives a posi- The research leading to these results has received fund- tive contribution at small wavenumbers. This is a strong ing from the French Agence Nationale de la Recherche signature of two-dimensional turbulence for which, con- (Programme Blanc ANR-12-BS09-011-04). versely to three dimensions, the nonlinear terms are not transferring kinetic energy toward small scales partici- pating to its accumulation at largest lengthscales of the flow. This term is exactly compensated by the linear Ekman friction and the coupling with the particle phase [1] W. W. Grabowski and L.-P. Wang, Ann. Rev. Fluid which are both negative and of the same order. Coupling Mech. 45, 293 (2013). is thus pumping energy at large scales but restitutes it [2] A. Johansen, M.-M. Mac Low, P. Lacerda, and M. Biz- zarro, Science Advances 1, e1500109 (2015). at larger wavenumbers as it is positive for k ≥ 4. In 112 the inertial range for 10 < k < 100 where both the con- [3] J. Bec, H. Homann, and S. S. Ray, Phys. Rev. Lett. , ⇠ ⇠ 184501 (2014). tribution of Ekman friction and viscous dissipation are [4] K. Gustavsson, S. Vajedi, and B. Mehlig, Phys. Rev. negligible, it is exactly compensated by a negative value Lett. 112, 214501 (2014). of the nonlinear transfer term. Both curves decrease as [5] P. Jenny, D. Roekaerts, and N. Beishuizen, Prog. Energy k−1, in agreement with the scaling observed earlier. At Combust. Sci. 38, 846 (2012). the smallest scales, coupling becomes negligible, nonlin- [6] J. F. Kok, E. J. Parteli, T. I. Michaels, and D. B. Karam, Rep. Prog. Phys. 75, 106901 (2012). ear transfer changes sign and is compensated by viscous 316 dissipation. The whole two-dimensional picture thus con- [7] A. W. Visser, Science , 838 (2007). [8] A. Johansen, J. S. Oishi, M.-M. Mac Low, H. Klahr, firms the prediction made above. T. Henning, and A. Youdin, Nature 448, 1022 (2007). We have thus evidenced from this work a new regime of [9] G. Boffetta, A. Celani, F. De Lillo, and S. Musacchio, turbulent flow where the feedback of suspended particles Europhys. Lett. 78, 14001 (2007). onto the fluid flow dominates inertial-range energy trans- [10] M. Wilkinson, B. Mehlig, and V. Bezuglyy, Phys. Rev. fers. This regime is evidenced by numerical simulations Lett. 97, 048501 (2006). 84 in two dimensions but such strong effects should also be [11] K. Gustavsson and B. Mehlig, Physical Review E , present in three dimensions, at least at sufficiently small 045304 (2011). [12] M. R. Maxey, J. Fluid Mech. 174, 441 (1987). scales. A remarkable feature of this turbulent enhance- [13] M. R. Maxey, Phys. Fluids 30, 1915 (1987). ment due to dust-like particles is the creation of small- [14] G. Haller and G. Yuan, Physica D 147, 352 (2000). − scale eddies whose spectral signature is a k 2 power-law [15] S. Chandrasekhar, Hydrodynamic and Hydrodynamic sta- range for the fluid velocity. These vortices profoundly bility (Oxford University Press, 1961). affect particle concentration. On the one-hand, their [16] U. Frisch, Turbulence: the legacy of A.N. Kolmogorov spatial distribution tends to weaken large-scale inhomo- (Cambridge University Press, 1995). geneities, to reduce potential barriers to transport and [17] E. Balkovsky, G. Falkovich, and A. Fouxon, Phys. Rev. Lett. 86, 2790 (2001). enhance mixing. On the other hand, the dispersion in the [18] J. Chun, D. L. Koch, S. L. Rani, A. Ahluwalia, and L. R. flow and the interactions between these long-living struc- Collins, J. Fluid Mech. 536, 219 (2005). tures trigger density fluctuations that are much more in- [19] F. Laenen, S. Musacchio, and J. Bec, “Modulation of tense than in the absence of coupling between the two two-dimensional turbulence by small heavy particles,” phases. Such effects clearly need being investigated in (2017), arXiv xxx. CHAPTER 6

Conclusions and perspectives

This thesis explored several problems related to the transport of particles by turbulent flows. While such problems occur in many natural and industrial situations ranging from combustion engines to planet formation, predictive models are mainly based on eddy- diffusivity approaches and cannot accurately handle high concentration fluctuations or non- trivial feedback of the dispersed phase on the fluid flow. We have focused here on these two aspects: The first part of this thesis was dedicated to the dispersion by turbulence of tracers continuously emitted from a point source. The second part concerned the introduction of a novel numerical method to simulate the transport of inertial particles and to understand how they modulate the carrier turbulent flow.

6.1 Turbulent transport of particles emitted from a point source

In chapter 3, the emission of tracers from a point source in the two-dimensional inverse cascade was studied. The main issue characterizing such a system is that, even if particles are released from the same spatial location, they enter the domain at different times. As a consequence, quantifying turbulent mixing requires understanding both the temporal and spatial correlations of the flow. Numerical simulations of the two-dimensional inverse tur- bulent cascade have been performed. Lagrangian tracers were seeded from a fixed point in space into an Eulerian velocity field integrated by pseudo-spectral methods. An analysis was carried of the mean displacement of tracers, as well as of the average radial concentra- tion profile. This allowed to conclude that the time-averaged dispersion is well described by two successive phases: A ballistic motion from the source with a characteristic velocity

113 114 CHAPTER 6. CONCLUSIONS AND PERSPECTIVES given by the large-scale flow, followed by diffusion after a time equal to the Lagrangian velocity correlation time.

Furthermore, a scaling analysis of the quasi-Lagrangian mass mQL as a function of the distance from the source has been performed. To this end, the fractal correlation dimension 2 was first measured using a classical box-counting algorithm at high spatial resolution Nx = 40962. The signature of the linear particle distribution, due to the injected mechanism, was found to persist at small scales. This is in qualitative agreement with the persistence of the inhomogeneity (due to the injection mechanism) at small scales as it was concluded in Celani et al. (2007) in the case of the Kraichnan model for the passive scalar. However, the correlation dimension of the tracer distribution showed to be largely contaminated by the recurrence of the Wiener process in two dimensions. Long-living trajectories indeed come back arbitrarily close to the source and infinitely often, contributing a uniform background to the concentration of tracers and thus acting as a source of homogenisation. This effect was more and more contaminating the statistics as the particles maximum lifetime was increased. To circumvent such difficulties, an alternative way to measure concentration fluctuations was further presented. This approach is based on a novel phenomenological description, which exploits the distortion of the emitted line as a function of time in order to quantify its contribution to the mass contained in balls of given size r and at a given distance R from the source. Although this approach is strictly valid in the limit of a continuous emitted line, some difficulties are encountered when working with discrete trajectories. In Lagrangian simulations, the continuity condition is quickly broken and we have shown that this effect limits the scaling analysis to a minimum size r that, in turn, depends on R. A natural extension of this work could consist in relating the quasi-Lagrangian mass scaling to the two-point equal-time correlation function. A comparison of this quantity in the case of the Kraichnan velocity ensemble (Celani et al., 2007) would allow one to stress the impact of a finite correlation time of the velocity. Also, as the Wiener process is not recurrent in dimension three and higher, it is expected that the determination of the correlation dimension D2 will not suffer from the homogenisation issue that has been encountered in two dimensions. It would be interesting to display the correspondences between the quasi-Lagrangian mass scaling obtained with the determination of D2 and the method we have presented by performing simulations in the three-dimensional direct cascade. Another point that would be worth studying further is the universal character of the mass scaling. Indeed, the Richardson super-diffusive regime is a consequence of the non- smoothness of the velocity in the inertial range. For a kinetic energy spectrum E(k) k−↵, / we have r2 t4/(3−↵). Other types of turbulence display the same scaling E(k) k−5/3, / / such as surface⌦ ↵ quasi-geostrophic turbulence (SQG), or single-layer QG model. This kind of turbulence is for example observed at ocean surface (Lapeyre & Klein, 2006) and upper troposphere (Tulloch & Smith, 2006). 6.2. MODELISATION OF SMALL INERTIAL PARTICLES 115

6.2 Modelisation of small inertial particles

In chapter 4, a numerical method has been proposed in order to resolve the kinetic equation associated with the dynamics of small inertial particles. This method allows to recover an Eulerian field for the particle density even at high inertia when the particle velocity dispersion has to be taken into account. Simulations were carried along with Lagrangian particles to assess for the validity of the method. In a one-dimensional random flow, the ability of the method to recover the phase-space fractal distribution was assessed through the study of the radial distribution function. In two dimensions, it was shown that this approach is able to reproduce the spatial distribution of the particles and their centrifugal ejection outside the vortical structures of the flow. Such preferential concentration effects were tested in two-dimensional turbulence, both in direct and inverse cascade, as well as in a cellular flow. The numerical convergence of the proposed method was studied via the analysis of the numerical error with respect to the velocity resolution. Also, the numerical performance was compared to Lagrangian simulations, showing an advantage for the Eulerian formulation as long as moderate errors O(10−2) on the density are tolerable. Among questions that remain open, let us mention possible improvements of this method. For example, one can wonder whether or not the designed approach would be adaptable to polydisperse suspensions. Up to now it was used to integrate numerically the Liouville equation for the particle density in the position-velocity phase space (x, u), focusing on particles that are characterised by a unique Stokes number quantifying their inertia. A naive way to consider N particle phases (of different sizes or masses) would be to implement N different density fields f. This would obviously multiply the computational complexity by N. This would open the way to consider interactions exchanging mass and momentum between the different solid phases, such as collisions, aggregation or coales- cence. This would amount to considering the density f in the phase-space (x, u,St). An extra-term in the Liouville equation would then appear of the form @St [G(St)f]. The ker- nel G(St) stands for how a specific phase is affected by the medium or possibly by the other phases, like an evaporation rate for example. The variable St is used here for generality but the volume v of the particle is often considered as the additional mesoscopic variable, as in the Williams equation (Williams, 1958) dealing with the issue of polydisperse sprays. Actually, approaches have been proposed to integrate this equation in the limit of small inertia, i.e. when velocity dispersion is negligible. Those originated in Tambour (1980) with the idea of sectioning the v dimension. Later on, this method has been named Eulerian multi-fluid methods. Each fluid is represented as a statistical average of the distribution in each volume section. This method has received later on multiple improvements (Laurent et al., 2004; Fox et al., 2008). The dynamic in the volume space can also be performed with a finite-volume method, and the issue of numerical diffusion is also present. Fox et al. (2008) have presented a method called direct quadrature method of moments that seems to reduce this spurious diffusion. In those cases, it was stressed that the performance of such 116 CHAPTER 6. CONCLUSIONS AND PERSPECTIVES

Eulerian descriptions was still competitive compared to Lagrangian simulations (Laurent & Massot, 2001), thanks to easier parallelisation and easier treatment of fragmentation and coalescence. However, when the phase-space dimension has to be further increased to account for velocity dispersion, the issue of competition with Lagrangian methods is not that trivial. Another extension of the proposed would be to use it for developing Large-Eddy Sim- ulation of particle suspensions. Basically, LES for the fluid phase consists in filtering the Navier-Stokes equations. The same technique applied to the kinetic equation yields:

u v v · ( uf u f ) @t f + x · (v f )+ v · h i− f = r h i−h ih i , (6.1) h i r h i r ✓ ⌧p h i◆ − ⌧p where denotes a spatial averaging (RANS method), or filtering (LES). The right-hand hi side denotes the interaction between particles and sub-grid scale fluid eddies and is the main object that needs to be modelled. For instance, Zaichik et al. (2009) have expressed this right-hand side in terms of a Gaussian integration by parts, yielding additional diffusive terms, but other closure approached can easily be tested. Figure 4.14 showed that the lattice particle method was able to reproduce the correct spatial density even when using 2 2 2 a resolution Nx = 512 against a resolution of 2048 gridpoints for the fluid velocity field. This is equivalent to considering a filtered velocity field in the kinetic equation (4.14) and constitutes a promising result in the framework of LES. It would be worth comparing the method we have developed in this thesis with LES approaches with modelled closures.

6.3 Turbulence modulation by small heavy particles

In chapter 5, numerical simulations have been performed in order to explore the effect of two-way coupling in a system with heavy particles in two dimensional direct cascade. This study has been carried out in the two asymptotics of small and large Stokes numbers, using two different models for the particle transport. Chapter 5 reproduces the scientific article that presents the main results from this work in the case of the Large Stokes numbers. In this asymptotics, the numerical method presented in 4 has been adapted in order to include the back-reaction from the particles onto the fluid. The Eulerian formulation of the particle population in the position space allows for an easier and more natural implementation of two-way coupling. In addition, no closing, model or reconstruction for the particle to fluid interaction is needed contrary to Lagrangian-Eulerian methods, although much progress has been made in the last few years in this domain (see, for instance, Gualtieri et al. (2015); Ayala et al. (2007)). Several statistical quantities have been measured for various coupling intensities, de- fined by the mass ratio φm. An effect directly observable when looking at the instantaneous density is the increase of particle clustering outside high-vorticity regions. The impact of two-way coupling on intermittency was also measured. In the two-dimensional direct en- strophy cascade, the distribution of vorticity gradients is characterised by broader tails than 6.3. TURBULENCE MODULATION BY SMALL HEAVY PARTICLES 117

Gaussian distributions. The action of the particles was shown to broaden this distribution, i.e. to increase its flatness, as the mass load ratio is increased. Gravity has been neglected in the dynamics we considered. This is a valid approxima- tion when fluid accelerations are stronger than g and when the fluid turbulent velocities are larger than the particle settling speed, which is more and more valid when the particles are not too massive. However, we presented an analysis with increasing mass load φm, i.e. with particles more and more massive with respect to the fluid. It would be interesting to include the gravity force to see how it affects the statistics we have presented. For example, experiments at large Stokes numbers carried by Hwang & Eaton (2006) showed that turbu- lent attenuation was uniform in the horizontal plane, in agreement with our measurements, and less pronounced in the vertical direction at high wavenumbers . Simulations have been performed in the two-dimensional direct enstrophy cascade. In −1/2 that case, a single time scale ⌧f = (2Z) can be defined (Z denotes here the average enstrophy of the fluid flow). However, in the inertial range of two-dimensional inverse energy cascade or three-dimensional direct cascade, the characteristic time of the flow −1/3 2/3 ⌧f = ✏ r depends on the observation scale r. So does the particles Stokes number −2/3 ⇤ with St(r)=⌧p/⌧f r which is a decreasing function of r. For a given r in the / ⇤ ⇤ inertial range, we may consider a time scale ⌧p such that St(r ) = 1. In this situation, the large scales r>r⇤ would be affected by low-inertia particles the small scales r

119

APPENDIX A

Software details

In these appendices, I discuss briefly about softwares that I have developed and used during my PhD thesis. The goal is to provide a transparent view to the reader about various numerical implementations and physical aspects.

A.1 GPU2DSOLVER

This C++-library was designed to solve Navier–Stokes equations in two dimensions in a periodic box.

A.1.1 Numerical scheme The library solves the incompressible two dimensional Navier–Stokes equations for the ? stream-function (x, y,t) from which the flow velocity is derived by u = r =(@y , −@x ). This allows one to integrate only one quantity instead of the two velocity components, which saves memory while ensuring incompressibility. The equation solved is equation (2.18), and is discretised in time using second order Runge–Kutta scheme. The numerical integration relies on the pseudo-spectral method, in which the solution is decomposed onto a Fourier basis, and the evolution equation is solved for the Fourier coefficients uˆ(k), with k the wave-number of the mode. The general advantage of the spectral methods (Orszag, 1969), especially involving Fourier decompositions suited for periodic boundary conditions, stands mainly in their exponential rate of convergence for smooth fields (faster than any polynomial in the grid size) (Bardos & Tadmor, 2015; Canuto et al., 2012). Furthermore, the evaluation of the derivative is much more precise than finite difference methods. Indeed, one does not need

121 122 APPENDIX A. SOFTWARE DETAILS to approximate derivatives using a finite number of points because it is straightforwardly given by the multiplication with the wave-number vector in the Fourier space. For instance: · u(x)= eik·xk · uˆ(k). r AnotherR advantage lies in the existence of fast algorithms to compute Fourier transfor- mations which only involve N log N operations,where N is the number of field elements, instead of N 2 as would perform a naive implementation (Walker, 1996; Brigham & Brigham, 1974). In addition, those algorithms were also been adapted and optimised for various par- allel architectures (for memory distributed cluster, general purpose graphical processing units, many-integrated core chips...). Spectral methods are also less expensive than finite element methods for simple open geometries. The term pseudo means that a part of the integration is done in the physical space. In- deed, fields products appearing in the Navier-Stokes equation (in the non-linear, convective term) translate into convolution in Fourier space involving N 2 operations. Transforming those fields in the physical space to preform this element-wise product and transforming them back again in the Fourier space only involves 4N log(N)+N operations, instead of N 2 for the convolution. The field transformed back into Fourier space must be de-aliased: the nonlinear term computation creates non-physical high-k modes. The classical 2/3 zero- padding rule is used, setting (k)= (k)Θ(k k ), i.e, (k)=0 k s.a. k >k , max −k k 8 k k max with kmax = N/3. In other words, all modes outside a circle of radius kmax are removed. The potential is initialized to (x) = 0 by default or may be read from an external file. This allows simulations to restart from a previous state, for example equilibrated flows in high resolutions. It may also be prescribed analytically to test for special solutions or to generate benchmark flows, like cellular ones (see section 4.5.1).

Viscous dissipation and hyper-viscosity

The integration of the dissipative term is made implicit:

t+∆t(1 + ⌫ k 2)= t. (A.1) k k

Hyper-viscosity is also implemented, for which viscous dissipation operator uses a higher power of the Laplacian: Dˆ = ⌫ 2q, yielding in Fourier space Dˆ = ⌫( 1)q+1k2q. This − r − − is often used to extend the inertial scaling range in the energy and enstrophy spectra and is especially suited when studying dynamics in this regime, such as the Richardson pair separation issue in the inverse energy cascade (see chapter 3). In this formulation for Dˆ, ⌫ has to be scaled accordingly and may go down to 10−47 for q = 16 (Smithr & Yakhot, 1994). To avoid having to renormalise the viscosity and use a value of the similar order for 2(q−1) ff ˆ 2 k di erent resolutions, we used for the operator the form D = ⌫k k . − ⇣ max ⌘ A.1. GPU2DSOLVER 123

A.1.2 Forcing Conservation equations for the fluid momentum writes:

@t(⇢f u)+ur · (⇢f u)=−rp + r · Σ + f. (A.2)

These equations are dissipative in the case of f = 0 due to the viscous forces in the constrain tensor Σ (see section 2.1), which physically result from small-scale friction between fluid particles. Starting from a non zero initial condition, energy would thus decrease until the flow is at rest (u = 0). Meanwhile, turbulence may arise due to non-linear transfer of energy through the wave-numbers (see 2.2.2). Studies of such non sustained systems in various situations, is called decaying turbulence in the literature. However, all experiments that I present in this manuscript involve flows in statistically steady state, hence which are sustained by an external source of energy. In real situations, such a forcing may come from mechanical agitation (propellers, high speed flows through a pipe...), or thermal effects (nuclear and chemical reactions in flows...). The numerical forcing that is implemented in GPU2DSOLVER is volumetric: momen- tum is added in Fourier space to modes in a given range of wave-numbers [kinf ,ksup]. For example, a forcing at large scales |k| is often associated to stirring. ⌧ Forcing methods is still a matter of debate depending on the applications. Indeed, although it is needed to reach a stationary steady state, one has to make sure that the forcing will not influence the statistics of the flow. For example, it is generally admitted that small scale dynamics are decoupled from large scale ones for sufficiently large Reynolds numbers (Eswaran & Pope, 1988). One can use multiple numerical methods (this list is not exhaustive):

• Freeze the forced modes to a constant value, fixing the large scale structures.

• Add a constant quantity to the forced modes: f = cst.

• Renormalise the total energy in the forced region so that at every time step such that: kend | |2(k)dk = cst. (A.3) Z kfirst

• Use a stochastic process for f. This one is widely used due to its randomness which is believed to be more realistic.

The forcing methods cited above have all been implemented in GPU2DSOLVER. In practice, only the last one, non deterministic, was retained in the studies I have carried, 124 APPENDIX A. SOFTWARE DETAILS mainly for the reason of being more physically realistic. Two random processes are imple- mented: for the first one, the force follows an Ornstein-Uhlenbeck process:

df(k)=−✓(f(k) − µ)dt + σdW (k). (A.4)

It consists of one drifting term, ✓µdt, one damping term, −✓f(k), and one diffusion term, 2 Gaussian, of variance σ . The processes are independant, i.e dW (k1)dW (k2) = δ(k1 h 2 i − k2). These processes have a non zero correlation time given by σ /2✓. µ is the asymptotic average value and is always set to 0 in the studies I have carried.

f(k,t)=f(k,t ∆t) exp( ✓∆t)+pσ∆t ⇠(k,t) (A.5) − − with ⇠ a number from the normal distribution N (0, 1). For more details about this process numerical integration, see Gillespie (1996); Honey- cutt (1992a,b). The other stochastic forcing implemented is a white noise:

f(k,t)=σ⇠(k,t) (A.6)

This kind of forcing is delta-correlated in time: f(t)f(t0) = δ(t t0) and is especially h i − suited for small scale forcing, as the correlation time of the velocity decreases with the scale. In practice, for both forcing types, σ is normalized in the following way:

σ0 = σ/(k2 2⇡(k k )(k + k )/2). (A.7) sup − inf sup inf The k2 factor removes the k2 dependence arising by the growing number of modes in shells between k and k + 1. 2⇡(k k ) accounts for the number of forced modes, and the sup − inf (ksup + kinf )/2 normalises by the middle kf . This normalisation allows one to use a forcing amplitude independent of the spectral forcing band width, the mean forced mode and the increase of the number of modes in each shell [k, k + 1]. Simple Euler temporal discretisation is used:

(k,t)= (k,t ∆t)+f(k,t). (A.8) − This is sufficient because we are not interested in a precise resolution of the forcing pro- cesses, as they are just random sequences.

A.1.3 Parallelisation The software makes exclusive use of general-purpose graphical programming units (GPGPU) to integrate the flow, using the proprietary CUDA C-interface (Nickolls et al., 2008). In- deed, this architecture was chosen because it is at present the best suited for massively A.1. GPU2DSOLVER 125 independent calculations, like in our case the update of all fluid Fourier modes, or lin- ear algebra operations. Furthermore, fast Fourier transforms also benefit from very well optimized algorithms for GPU architectures. In my case, use was made of the cuFFT library. The cuRAND library was also used to generate pseudo-random sequences for the forcing. One sequence was generated for each forced mode in a way that allows one to simulate reproducible flows. A quick benchmark with a pseudo-spectral 2D solver showed that simulations using one single graphic card perform as well as distributed memory sim- ulations (using message passing interface) with 120 cores. Nevertheless, the choice of ⇠ this proprietary language implies that the software is only capable to run on systems with CUDA-capable graphic cards. It should be stressed that the very limiting factor for high resolution fluid simulation is the available memory. The GPUs I have used during my thesis were middle-range Tesla M2050 graphic cards from Nvidia which had a video memory of 3.5GB. I was able to 2 2 ⇠ perform fluid computations with resolutions up to Nx = 8192 , which only corresponds 3 3 to Nx 400 in three dimensions while state of the art three dimensional computations ⇠ 3 3 go up to Nx = 8192 especially in cosmological simulations (Yeung et al., 2015). Newer generations of graphical processing units can go up to O(10GB), but it is still limited to perform high resolution three dimensional simulations on a single graphic card. It is thus necessary to couple those acceleration devices with distributed memory architectures using message passing protocols among many computational nodes. The performance then hinders from inter-node communications. Nowadays high performance computing codes show to linearly scale up to O(105) processors for O(61443) resolutions (see, for instance, Mininni et al. (2011)). Figure A.1 displays a benchmark to assess the performance of the software. Number of points integrating by a full Runge-Kutta 2 step per second is shown as a function of the resolution Nx. It corresponds to the the number of evolution of the full 2D field multiplied 2 by Nx . This quantity was computed by averaging over a fixed number of 600 time steps. 0 For a single CPU, it would behave as Nx , i.e. constant. In MPI implementations, benchmarks are often presented as the time (in seconds or minutes) per time step as a function of the number of processor for a fixed resolution. Performance generally scales linearly with the number of processor until it eventually reaches a plateau, displaying the bottleneck of messages communication which limits the performance. The metric displayed in Figure A.1 is rather analogous, the constant behaviour being the proof of the absence of an additional limiting bottleneck. Notice that in this case, as a single GPU card is used in this case, the number of available threads and cores is fixed. A slope greater than 0 indicates that the scaling is better than a single-threaded ex- ecution, and this is what is indeed observed for small Nx. This can be explained by an increase in the device occupancy: increasing the resolution, hence the number of points to be treated, exposes more parallelism, i.e. there are less idle threads and the GPU card is more efficiently exploited. The saw-tooth like behaviour is due to the alternating resolution between Nx being or 126 APPENDIX A. SOFTWARE DETAILS

109 Simple precision Double precision Npoint/sec = Cst. sec / 108 point N

107 103 104 Nx

Figure A.1: Benchmark of the GPU2DSOLVER software performed on a middle-range GPGPU, the Nvidia Tesla M2050. The number of points integrated per second is repre- sented as a function of the resolution Nx up to Nx = 8192. not a pure power of 2. FFTs algorithms perform better for combinations of prime number a b c d powers, i.e. 2 3 5 7 ..., the lower the number the better. In this case, Nx is alternatively 2i and 2i−13, with i =8, ..., 13, excepting 2123. This code is under CECILL-C license and available on demand.

A.2 LAGSRC2D

This C++ library was designed to simulate emission of particles from a punctual source and removal from the domain following a custom criterion. The particle system may then be prescribed an external fluid velocity for its transport. The work in which it was used can be found in chapter 3.

A.2.1 Numerical implementation A class is built to represent the whole particle system. Because it is generally more optimal to pack in memory contiguous data, mainly to facilitate cache reuse (Cheng et al., 2014), it was decided to encompass in it the arrays representing particle properties, i.e. positions, velocities, accelerations, unique identifiers and ages. Each property is a GPU array, more precisely Thrust device vectors, from the Thrust library distributed with the CUDA toolkit (Bell & Hoberock, 2011). This library offers the advantages of transparent arrays utilisation A.2. LAGSRC2D 127 and C++ standard template library (STL)-style high level algorithms. Another way of designing the system would be to use fixed-length arrays, with a size large enough to contain the desired maximum number of particles present in the system. Then when particles exit the system due to some boundary conditions, they are reinjected. This solution allows a gain in performance, because less time is spent to reorganize memory by removing and adding elements at random places in the vectors. However, it would create spurious temporal correlations between the injection intensity and boundary conditions. Regarding the physical issue for which the library was designed, this is an undesired effect. The design of this class was part of the motivation to work on the SoAx library project (see appendix B). Indeed, this last library allows one to manipulate a particle system and, through macro-generated function, to add very simply properties against which class methods can be called. LAGSRC2D is able to handle two types of removal conditions: particles are removed after an age Amax or after a maximum distance from the injecting source. At a frequency chosen by the user, the system checks for the exit condition and erases elements for which this condition is true. This is done through a call to C++ standard library functions remove if and erase, which gives a linear complexity of N comparisons and some n deletions, with generally n N . ⌧ The dynamical update of particles position and velocities make use of the second order Adams-Bashforth scheme (Jeffreys & Jeffreys, 1999):

3 1 vt+∆t = vt + ∆t( At At−∆t)+p2∆tσ⇠(t). (A.9) i i 2 i − 2 ∆t is the time step, At stands for the i component of the acceleration at time t, ⇠ a random i − number from N (0, 1) distribution and σ the noise amplitude parameter to add a random perturbation. This perturbation typically results from random shocks with fluid particles. In chapter 3, σ = 0. When the knowledge of the flow velocity at the position of the particle u(Xp) is required, its value is determined by a bicubic interpolation, thus involving 16 points around the particle. Files for chosen properties are written on the disk with the time of the output and a restart is then possible from any of those output by specifying the restart time.

A.2.2 Injection rate A parameter N , called injection intensity is used to control the number of particles I ⇤ emitted in a duration of t = 1. Another parameter determines how often the injection takes place, the injection period T . The number of particles to be emitted during each period is then N ⇤ = N ⇤/T , which may be a floating point number. IT I Then, to ensure that an integer number of particles is injected per period, N ⇤ is IT adjusted (simply rounded) so that the computed NI may be a bit higher or lower than 128 APPENDIX A. SOFTWARE DETAILS requested. This number may then be zero, and if so the user is issued a warning and has to ⇤ modify the injection period. The maximum deviation expected between required NI and actually computed N is thus |N − N ⇤|/N ⇤ = 1 t/T . In practice, injection is done at the I I I I 2 NI ⇤ beginning of each period. This code is under CECILL-C license and available on demand.

A.3 LOCA: Lattice model for heavy particles

This C++ library was designed to simulate transport of heavy particles in externally- prescribed flows. The challenge is to take into account the particles velocity dispersion at a given point in space. For more details on this physical issue, see chapter 4. As explained in chapter 4, the equation (4.14), which is a transport equation in the position-velocity-phase-space for a scalar quantity, is to be discretised and represented in 4 dimensions.

A.3.1 Finite volume fluxes Inertial particles heavier than the fluid cluster in regions of low fluid vorticity. Initial uniform distributions will thus develop strong density fronts in the position space (see for example section 4.5.2), and also in the two dimensional velocity space, where a finite volume scheme is used for the numerical integration. Indeed, particles tend to relax toward the fluid local velocity u(x) with a characteristic time ⌧p. The distribution width around this velocity is thus an increasing function of ⌧ , and when ⌧ 0, particles perfectly align p p ! their velocity to the fluid one, corresponding to a δ-function in the velocity space. The finite volume should then be able to adapt to strong gradients when necessary. Furthermore, positivity preservation is a necessary condition when dealing with positive quantities such as mass, or substance concentration, as it is often the case in engineering simulations. This property can be ensured requiring the TVD (total variation diminish- ing) property of the scheme, which ensures not to create new extrema in the function. I thus needed a finite volume discretisation which gave both good accuracy, that could automatically adapt to strong fronts, and that preserved positivity. The natural solution is the use of flux limiters, which are function of the local gradients, either flux or state gradients. They are used to tune the part of the flux that comes from a high order scheme and the one that comes from a low order scheme. A lot of limiters exist, and after some trials among a few other TVD ones, the one from Koren (1993) was chosen. It offered the best accuracy, although only the test for one spatial resolution in the one dimensional case (Nx = 2048) was performed. Its expression writes K(r) = max[0, min(2r, (2 + r)/3, 2)], where r is the local gradient. In the case of a positive net force, F>0 at the interface between cell i and i + 1 (the upwind direction), f f r = i − i−1 . (A.10) f f i−1 − i−2 A.3. LOCA: LATTICE MODEL FOR HEAVY PARTICLES 129

Flux at this cell interface is then computed by (Hundsdorfer et al., 1995):

1 φi+ 1 = fiFi + K(r)(Fifi − Fi−1fi−1). (A.11) 2 2 The density field is updated via a Euler temporal discretisation:

t+∆t,⇤ t ∆t fi = fi + (φi+ 1 − φi− 1 ). (A.12) ∆v 2 2 When including diffusion, the fluxes at the cell interface are applied subsequently:

f t+∆t,⇤ − f t+∆t,⇤ D i+1 i 1 ⇣ ⌘ φi+ = , (A.13) 2 ∆v ∆ t+∆t t+∆t,⇤  t D D f = f + φ 1 − φ 1 . (A.14) i i ∆ i+ i− v ⇣ 2 2 ⌘ A.3.2 Dynamic grid resizing (DGR) The domain grid in the velocity space consists of regular square cells disposed in the interval d [−Vmax, +Vmax] with d the dimension. Fixing this parameter forces the user to know a priori the value of the particles rms-velocity needed to be resolved. It can be estimated in some cases, as it was in the study discussed in chapter 4. In this case, some pre-run with Lagrangian particles were carried to estimate their rms-velocity, and this property was known to decrease with St1/2 (St is the Stokes number of the particles). Of course this knowledge is not always guaranteed. In particular, this is a problem faced for the study described at chapter 5. In that case, the flow exerts drags on the particles and the particles react back on the fluid. The system is thus coupled and it was a quantity of interest to determine how this velocity varied with the intensity of the coupling (defined via the mass ratio between the particle population and the fluid total mass). A tendency which could also depend on the particles inertia. Furthermore, the importance to provide a good value for the parameter Vmax is shown in section 4.4.2, where its impact on the error on the density field was assessed. It was thus decided to implement a dynamic re-gridding to be able to dynamically adjust the parameter Vmax depending on the particle dynamics. It is important to note that this method is different from adaptative mesh refinement, or AMR. This last technique is used in the scientific community for methods generating more densely gridded parts of the domain. It is widely used for large scale computing, for example in astrophysics (Teyssier, 2002; Mignone et al., 2011; Bryan et al., 2014), using a tree-based decomposition, advantageous for systems where small regions need large refinement. In our case, the number of cells is constant, and the 2d grid remains uniform. Only the parameter Vmax is adapted, so that the dynamic of the particles is better resolved. 130 APPENDIX A. SOFTWARE DETAILS

Figure A.2: Particle density field without (left) and with (center) DGR. Left simulation uses a fixed value of V = 4 while the real value is 3. Right simulation starts with max ⇠ Vmax = 4, resizing the velocity grid every 10⌧p at Vmax/vprms = 2. It resolves better the mass voids due to high vorticity regions (corresponding vorticity field shown on the right). Resolution is N 2 N 2 = 10242 132 and St 1. x ⇥ v ⇥ ⇠

This adjustment takes place at a frequency specified by the user, and the desired ratio

Vmax/vprms is also prescribed. At each adjustment, the resolution in the velocity space ∆v = 2Vmax is modified and so is the time step ∆t = ∆x/∆v which is a necessary criterion Nv for agreement with the discrete displacements on a lattice (see section 4). The transfor- mation between the particle population before DGR, f old, and after DGR, f new is made new old through a bilinear interpolation in the velocity space. If Vmax

Of course, one has to provide a minimum Vmax. Otherwise, ∆v would tend to 0 and the stability criterion would not be fulfilled any more (see section 4.3 for more details on stability). This is particularly the case when starting from rest initial condition.

This technique was tested on a benchmark with a two dimensional direct enstrophy cascade at resolution N 2 N 2 = 10242 132. Two simulations are performed using the x ⇥ v ⇥ same stochastic forcing, with and without DGR. Figure A.2 displays two instantaneous density fields for St 1. One can see that DGR allows to reduce numerical diffusion and ⇠ to better resolved the clustered trajectories outside the fluid vortices. Systematic study of the impact of this grid resizing would require to assess the numerical error between Lagrangian particles and corresponding reconstructed Eulerian mass fields, as a function of the ratio Vmax/vprms = 2 and the frequency of adjustment. A.3. LOCA: LATTICE MODEL FOR HEAVY PARTICLES 131

A.3.3 Parallelisation Although the flow is computed using GPU parallelisation, the lattice method was integrated on the CPU using shared memory parallelisation with the OPEN-MP library. The overhead of fetching the fluid data located on the GPU was compensated by the fact that the time step of the flow was always smaller than the time step of the lattice particles, allowing for a not so frequent memory fetch. It was indeed verified through profiling that the part of the computation time dedicated to the memory transfers between CPU and GPU was negligible (O(1%)). Furthermore, this allows to release pressure on the scarcely available memory on the GPU, and to perform CPU computations in parallel to the GPU. Finally, the algorithm was performing better using shared memory than GPU accel- eration. This is easily interpretable: GPU acceleration functions better when processing adjacent memory locations. Because the full particle distribution f is stored in a linear array in memory with the velocity being the fastest varying dimension, the GPU paralleli- sation is efficient during the acceleration step, but certainly not during the streaming step, where lots of memory movements have to be performed by lightweight threads. Actually, this issue is common with the Lattice-Boltzmann GPU acceleration, for which efficient algorithms have been proposed (see, for instance, Habich et al. (2011)). This code is under CECILL license and available on demand. 132 APPENDIX A. SOFTWARE DETAILS APPENDIX B

SoAx: a convenient and efficient C++ library to handle simulation of heterogeneous particles in parallel architectures

A large variety of natural and industrial applications require scientists and engineers to use a collection of a large number of objects encompassing multiple attributes. For example, in the research framework of this thesis, these objects are solid particles suspended in turbulent flows and the attributes are, or can be, the positions, velocities, mass, electric charge, concentration, lifetime, etc. When designing a code to handle these attributes, it quickly appears that adding more properties becomes cumbersome in terms of code maintenance and reusability. Indeed, for performance reasons, it is desirable that each attribute be declared as a separate, linear array in memory, promoting cache reuse. The trace of this array has then to be kept and propagated in all the members of the class representing the particles. Adding a new attribute forces then one to pass in review all the methods of the class encompassing these arrays. The SoAx library basically solves this problem. It is placed under GPLv3.0 license. Although the chained-list model may be more advantageous in some situations, it is not when repeating the same operation for each attribute on every particles and repeatedly a large number of times as in numerical integrations of differential equations. This is called number crunching. A list of its capabilities are:

• Macro generated class members. Each attribute may be given a handy name, and each additional one is automatically added to member functions.

• GPGPU support. Users may choose that their data reside on a GPU device. In that case, all commands and computations are executed on the device. Transfer between

133 134 APPENDIX B. SOAX

CPU and GPU only occurs when the user explicitly retrieves the data from the GPU.

• MIC support.

• Expression templates: naive implementations of chained operations (for example: a = (b+c)*d/(e-f)) between arrays of data results in multiple traversal of the data, operations between executed between pairs of arrays, thus leading to poor cache utilisation. Expression templates, that arose since the great possibilities offered by template metaprogramming in C++, allow to embed the succession of operations to be performed an to apply them in a single traversal of the data (see, for instance, Iglberger et al. (2012)).

• MPI-ready. Particles may be individually retrieved, shared among processors and saved to .

Performances of SoAx are measured as a function of the number particles on several architectures (MIC, GPU). For each case, they are compared to a classical implementation, i.e. not using macro generated class members or expression templates, to assess for the validity of the implementation. The paper issued from this development submitted to Computer physics communica- tions is reproduced hereafter. SoAx: A generic C++ Structure of Arrays for handling particles in HPC codes

Holger Homann∗, Francois Laenen Laboratoire J.-L. Lagrange, Universit´eCˆoted’Azur, Observatoire de la Cˆoted’Azur, CNRS, F-06304 Nice, France

Abstract The numerical study of physical problems often require integrating the dynamics of a large number of particles evolving according to a given set of equations. Particles are characterized by the information they are carrying such as an identity, a position other. There are generally speaking two different possibilities for handling particles in high performance computing (HPC) codes. The concept of an Array of Structures (AoS) is in the spirit of the object-oriented programming (OOP) paradigm in that the particle information is implemented as a structure. Here, an object (realization of the structure) represents one particle and a set of many particles is stored in an array. In contrast, using the concept of a Structure of Arrays (SoA), a single structure holds several arrays each representing one property (such as the identity) of the whole set of particles. The AoS approach is often implemented in HPC codes due to its handiness and flexibility. For a class of problems, however, it is know that the performance of SoA is much better than that of AoS. We confirm this observation for our particle problem. Using a benchmark we show that on modern Intel Xeon processors the SoA implementation is typically several times faster than the AoS one. On Intel’s MIC co-processors the performance gap even attains a factor of ten. The same is true for GPU computing, using both computational and multi-purpose GPUs. Combining performance and handiness, we present the library SoAx that has optimal performance (on CPUs, MICs, and GPUs) while providing the same handiness as AoS. For this, SoAx uses modern C++ design techniques such template meta programming that allows to automatically generate code for user defined heterogeneous data structures. Keywords: keyword1; keyword2; keyword3; etc.

PROGRAM SUMMARY/NEW VERSION PROGRAM Reasons for the new version:* SUMMARY Manuscript Title:SoAx: A generic C++ Structure of Arrays for Summary of revisions:* handling particles in HPC codes Authors: Holger Homann, Francois Laenen Restrictions: Program Title: SoAx Journal Reference: Unusual features: Catalogue identifier: Licensing provisions: Additional comments: Programming language:C++ Computer: CPUs, GPUs Running time: Operating system: Linux RAM: bytes Number of processors used: Supplementary material: Keywords: Keyword one, Keyword two, Keyword three, etc. References Classification: [1] Reference 1 External routines/libraries: [2] Reference 2 Subprograms used: [3] Reference 3 Catalogue identifier of previous version:* * Items marked with an asterisk are only required for new versions of Journal reference of previous version:* programs previously published in the CPC Program Library. Does the new version supersede the previous version?:* Nature of problem:

Solution method: 1. Introduction

Particles are at the heart of many astrophysical, environmen- ∗Corresponding author. tal or industrial problems ranging from the dynamics of galax- E-mail address: [email protected] ies over sandstorms to combustion in diesel engines. Investi-

Preprint submitted to Computer Physics Communications September 23, 2016 gating such problems require generally integrating the dynam- public : ics of a large number of particles evolving according to a given int∗ id ; physical laws. Examples are N-body simulations in cosmology double∗ position ; [1], particle in cell codes (PIC) exploring plasma physics [2] float∗ mass ; or hydrodynamic simulations studying Lagrangian turbulence } problems [3]. Such kind of numerical simulations have in com- It is then convenient to add member functions to this class mon that that they are numerically expensive meaning that they that perform operations on all the properties such as allocating rely on number crunching, i.e. an enormous number of float- memory: ing point operations. Studying the particle dynamics during a finite time interval requires the numerical integration of the un- Listing 3: Member function to allocate memory for particle property arrays derlying equations of motion over many time steps so that the void PartArr :: allocate ( int n) particle data (position, velocity, ...) is used in simple but numer- { ous repeated operations. The performance of such operations number = new int [n]; position = new double [n]; depend in a crucial way on how the particle data is stored and mass = new float [n]; accessed. } Modern supercomputers are often indispensable for studying challenging problems. Their architecture got more and more In the same way, member functions for adding, removing and complex in recent years. The today’s fastest computers consists other functionalities could be added. This kind of implemen- of several performance sensible components such as multi-level tation is called structure of arrays (SoA) from the fact that in caches, vector units based on the ’single-instruction multiple- this case one structure handles a set of particles whose proper- data’ (SIMD) concept, multi-core processors, many-core (MIC) ties are represented by different arrays. PartArr pArr; cre- and GPU accelerators. Evidently it is important to make use of ates a set of particles and an individual particle is referenced all these components to optimize the performance of a numeri- by the array index (pArr.position[42] returns the position cal code. of particle 42). A priory, particles cannot be extracted as indi- Particles can carry different properties such as an identity, a vidual objects from the structure PartArr. For this, a structure position or a mass. In programming languages such as Fortran, Particle (see codelet above) would be needed together with a C or C++, the data types int, double and float, could be function copying the array data for one index to the Particle chosen to represent the former particle properties. (In this paper member variables. From these considerations it is clear that codelets (serving as implementation examples) will always be AoSs are easier to implement and to use than SoAs. given in C++, but the reasoning will be kept general so that it AoS are also more extendable than AoS. Imagine one would will similarly apply to Fortran and C.) like to reuse the above outlined particle storage implementa- In C++, particles can be implemented as a heterogeneous tion for a slightly different particle type that requires the ad- structure dition of a property such as a charge. AoS are more flexi- ble than SoA for this task: the novel property could be added Listing 1: Particle structure storing the data of one particle to the Particle structure by simply adding the member vari- struct Particle { able float charge;. In the case of a SoA an array (through int id ; float* charge;) could be added to PartArr. But in turn, double position ; all member functions such as allocate would also have to be float mass ; updated in order to treat the added array. } AoS seem to be the better candidate to store particles than This way, individual particles can easily be generated as ob- SoA. However, SoA are faster in many circumstances (espe- jects (Particle p;) and modified (p.id = 42). A set of cially on MIC and GPUs) [4, 5] than AoS and we show that this particles is then often handled by an array- or list-like struc- is also the case for typical manipulations (such as trajectory in- tures (std::list pList;) providing functional- tegration) on particle data. By means of a benchmark modeling ities such as access, adding and removal of particles. Such an floating-point operations used in real codes we show that SoAs organization is called array of structures (AoS) as the particles are typically several times faster than AoSs and that the perfor- are represented by a structure that is hold by an array (or list). mance of an AoS depends on the size (in terms of bytes) of the Treating particles as objects is also convenient for transferring structure (Particle in the example above). In order to cope them from one process to another via the message passing in- with the seemingly contradicting properties handiness, flexibil- terface (MPI) in parallel applications. ity and performance, we present a generic implementation of a Another implementation strategy for handling a set of many SoA called SoAx that has optimal performance while providing particles is to use one structure that holds several arrays; one the same handiness and flexibility of an AoS. array for each particle property: This paper is organized as follows. In section 2 we bench- mark the performance of AoS and SoA on CPUs. In section Listing 2: Structure of array containing one array per particle property 3 we discuss a similar benchmark on GPUs and MICs. The class PartArr generic C++ implementation of SoAx is presented in section 4. { Conclusions are drawn in 5. 2 2. Benchmarking AoS and SoA on CPUs

In order to compare the performance of SoA and AoS we 0 32 measure the execution time of a benchmark computation. The 32 latter consists in performing an Euler advection time step for the position x of a set of particles

x+=dt v, (1) dt denoting the time step (a floating point number) and v the velocity of the particle. This equation is a simple prototype for typical operations appearing in numerical codes. It consists, for each component, of two loads from the heap memory plus one for the constant dt, usually from the stack, and two stores in heap memory. For benchmarking AoS we use the structure Figure 1: Benchmark comparing the performance of a structure of array (SoA) Listing 4: Particle structure used in the benchmark. SIZE is the number of and an array of structure (AoS). The index SIZE in AoSSIZE denotes the number supplementary floats. of supplementary floats in the structure Particle (List. 4). template< int SIZE> class Particle {

public : 0 float x[3]; 32 float v[3]; 32 float temp [SIZE ]; } ; where temp is a place holder for additional particle properties that might be necessary for the physical problem under consid- eration (such as a mass, an electric charge...) or the numeri- cal algorithm (such as temporary positions and velocities for a Runge-Kutta scheme). In the case of SoA we simply use three heap-allocated C++ arrays for x and v, respectively. Typically, in numerical simulations many successive time steps are performed in order to integrate the particle dynam- ics. In our benchmark we therefore loop many times over the Figure 2: Execution time of an array of structure (AoS) relative to that of a structure of array (SoA). The index SIZE in AoSSIZE denotes the number of numerical implementation of (1). We use standard compilers supplementary floats in the structure Particle (List. 4). with enabled optimization. Figure 1 compares the normalized execution time for the SoA and AoS as a function of the particle number. The SoA is An important drawback of AoS is that its performance de- much faster than the AoS. Their relative performance is shown pends on the size of the particle structure. The more data (prop- in Fig. 2. The SoA implementation is up to 25 times faster than erties) this structure holds, that is to say the bigger it is, the more the AoS and one gains at least a factor of two to three by using it fills the cache that in turn hinders performance. A particle a SoA instead of an AoS. with 32 additional floating point member variables (SIZE=32 The measured performance depends on the number of par- List. 4) in is much slower than its slim counterpart. This prob- ticles which is a consequence of the different cache levels of lem is of course absent for SoA as all arrays are allocated indi- modern CPUs. Usually they provide three levels with sizes of vidually and continuously in memory. Data (particle properties) 32 kByte (L1), 256 kByte (L2), and 8-40 MByte (L3). The that is not used in the execution loop will not be loaded into the colored arrows in Fig. 1 show the cache limits in terms of a cache. the number of particles of a certain size (in terms of bytes). The execution time of the AoS also depends on the container One observes that the performance is maximal when the L1 used to store the particle objects. A stl vector is significantly cache is filled and all particle data still fits into the L2 cache. faster than than a stl list. We measure roughly a factor of two. When the particle data size exceeds the L2 cache, the execu- This difference is due to the additional indirections involved for tion time slightly increases. An important performance drop linked lists (such as the stl list). On the other hand, a list is happens when data becomes larger than the L3 level. At that faster in removing particles than a vector as the latter copies point data has to be transferred from the main memory that has successive elements to keep the data continuous in memory. a significantly lower bandwidth than the caches. This drawback can be overcome when the ordering of parti- 3 cle is not important. In that case, a particle can be removed by loaded from the main memory which is too slow to efficiently simply overwriting it with the last particle. This strategy is used fill the vector registers. by default by SoAx. Vectorization does not speed up AoS computations. Appar- The performance measured with a given benchmark naturally ently, the auto-vectorizer of the compiler does not manage to depends on the architecture of the CPU. However, it is impor- create a substantial gain if a AoS is used. This means that a part tant to note that the just discussed relative performance (SoA vs. of the SoA superiority can be explained by the fact that SoA AoS) will not or only weakly depend on the clock speed. But effectively use the CPU vector units. other differences, especially the vectorization units are impor- tant as we will show now. We will consider two different CPU 5 architectures distinguished by the date of their commercial re- 4.5 0 lease. This sheds light on how the ’SoA vs AoS’ performance 32 ratio changed over time. We compare the SoA performance to 4 the maximal AoS performance (using the smallest possible par- 3.5 ticle size together with a stl vector). In Fig. 3 we compare Xeon 3 CPUs from 2010 and 2014. For the two CPU generations SoA clearly wins over AoS. But the modern chip has a higher perfor- 2.5 mance gain. Over only four years the gain has nearly doubled. 2

The CPU architecture is more and more constructed in a way 1.5 that favors the SoA layout. 1

0.5 10 100 1000 10000 100000 1 ⇥ 106 1 ⇥ 107 1 ⇥ 108

Figure 4: Performance gain due to vectorization for a SoA and AoSs.

This also explains the observed differences between the two CPU architectures. From one CPU generation to the other, the register width and the set of instruction has been augmented. The old CPU from 2010 has 128 bit vector register with a SSE4.2 instruction set and the most recent CPU from 2014 has a 256 bit vector register with an AVX2 instruction set. The factor of two between the 128 bit and 256 bit register explains the dif- ferences in Fig.3 for intermediate particle numbers. Of course other features than the vector unit changed among CPU archi- tectures but it seems that most of the changes in the ’SoA vs Figure 3: Benchmark comparing SoA and AoS for different CPU generations AoS’ performance ratio over the years are due to optimizations distinguished by the date of their commercial launch. 2014: Intel Xeon E5- of the vector units. 2680 v3 (Haswell EP); 2010: Intel Xeon X5650 (Westmere EP)

One architectural component that has changed over the years 3. Benchmarks on MICs and GPUs is the performance of the vector unit. All today’s CPUs possess so-called single instruction multiple data (SIMD) register and Today’s supercomputer often use accelerators to speed up associated instruction sets. These allow to perform the same computationally intensive parts of numerical codes. Mainly two instruction (such as an addition) to many floating-point number different accelerator types exist: at a time (in one cycle) that can significantly speed up code. In Intel recently introduced the ’many integrated core’ (MIC) Fig. 4 we compare the performance of SoA and AoS with and concept with the Xeon Phi co-processor that assembles many without the use of the vector unit. The vectorization gain of a computing cores (around 60) on one chip. The used computing SoA reaches four to five for small particle numbers of the or- cores are simplified versions of commonly used CPUs so that der of 100-1000 particles. The theoretical gain is eight as the numerical code compile without changes on a Xeon Phi. used CPU has a 256 bit vector register containing eight single Nvidia and AMD/ATI developed graphics processing units precision floating point values. At intermediate particle num- (GPU) that are now often used in high performance comput- bers (103-106) the gain is around two and vanishes for higher ing. This architecture uses hundreds to thousands of very sim- particle numbers. The origin of these regimes can be found in ple computing cores to speed up high parallel algorithms. For the three cache levels: The gain is maximal if all data fits into these GPUs the numerical code has to be especially designed. the L2 cache. The second regime corresponds to data fitting The importance of these accelerators for HPC is under- into the L3 cache. However, when the data size exceeds the lined by the fact that they are massively employed by the two latter the vectorization gain vanishes because the data has to be fastest supercomputers in the world (according to the TOP 500 4 list, www.top500.org). In fact, Tianhe-2 uses Xeon Phi co- processors and Titan Nvidia GPUs.

3.1. MIC During the last decade, the performance of supercomputers grew essentially by increasing the number of (standard) com- puting cores so that high performance computing demanded more and more for parallel numerical algorithms and codes. In- tel pushes now further in the direction of massive parallel pro- gramming by introducing co-processors, called Xeon Phi, with around 60 integrated cores each. A single core is in general compatible to standard CPUs but exhibits some architectural differences that are important for the performance of SoAs and AoSs: A Xeon Phi has no L3 cache but only a 32 kByte L1 and a 512 kByte L2 cache per core. Another aspect is that the vectorization capacities have been improved by extending the Figure 6: Speed-up by vectorization on a Xeon Phi SIMD registers to 512 bits which means that either 16 single precision floating point number or 8 double precision number can be processed in one cycle. providing a many-core device, separated from the CPU, and These design differences show up in the relative performance typically connected to this one via a PCIe band. Graphic cards of AoS compared to SoA (as before, we will only study the are widely used as accelerators in computer clusters, and power single-core performance). Our benchmark shows the the MIC many of the TOP500 fastest supercomputer. cores favor SoAs over AoS and that even more than standard A few thousand of threads can run concurrently on the CPUs. For small size objects and intermediate particle num- graphic card, thus providing the possibility to process many el- bers the tested SoA is roughly ten times faster than the AoS ements at a time. Furthermore, the architecture, labeled SIMT (see Fig. 5). If the stored particle has a considerable size, this (for Single Instruction, Multiple Thread) is somewhat different difference even varies between twenty and forty. from the SIMD in that every single thread has its own regis- ter state and can have independent behaviors from the others, a 45 feature allowing a thread-based as well as coordinated threads

40 0 development. 32 Another important difference from the CPU is the role of the 35 L1 cache. Different caches co-exist, each one belonging to a 30 given streaming multiprocessor, a structure responsible to dis- 25 patch the work among the threads. This cache is mainly used for register spilling and some stack variables. It does not pro- 20 mote temporal locality so that repeated operations on the same 15 memory locations will not necessarily benefit from this cache. 10 The L2 cache, shared among all streaming multiprocessors, will 5 be used instead. We thus expect the SoA pattern not to bene- fit from the L1 cache, but the AoS will in fact benefit from it : 0 10 100 1000 10000 100000 1 ⇥ 106 1 ⇥ 107 1 ⇥ 108 indeed, loading a large structure into memory allows threads to reuse close memory. A benchmark similar to those listed above is performed. The Figure 5: Benchmark comparing the performance of AoS to SoA on a Xeon graphic card used is a Nvidia Tesla M2050, a middle-range, Phi. widespread computing device. The card has 448 cores, spread among 14 multiprocessors and the L2 cache size is ∼ 786 The reason is the extended vector performance of the MIC kBytes. In the SOA algorithm, three functions are launched, cores. Up to the point when the L2 cache is filled, vectorization one per position and velocity component, with a number of speeds up the computation by a factor of roughly ten (see Fig.6) threads such that each thread has a single element to process. which is below the optimal value of sixteen but twice the speed- The program is compiled with optimization. Timing is mea- up measured for a standard CPU. Again, the cache size limits sured by the Nvidia profiling tool, allowing to isolate the kernel the particle number range for this speed-up. execution time from the overhead of the function calls. Execu- tion times normalized by the number of particles are shown in 3.2. GPU Fig. 7. For large particle numbers, SoA outperforms AoS solu- The architecture behind the General Purpose Graphical Pro- tion, by a factor ∼ 2 for S IZE = 0 and ∼ 20 for S IZE = 32. cessing Units (GPU) uses a divide and conquer philosophy, by As the particle number decreases however, AoS performs bet- 5 ter, with higher crossover for lower S IZE. The reason for this 10-6 lies in the GPU architecture, as we will now explain. AoS0 Figure 9 displays two relevant measured metrics for the func- AoS32 SoA tion used. The major drawback of the AoS approach is the well 10-7 know effect of uncoalesced memory access, hence threads fetch unneeded data in the cache lines. This is particularly damage- able in the case of GPU computing because the major weak -8 point is the latency of memory access. Accessing data is done 10 by a single, indivisible group of 32 threads, called a warp. Loading a large structure in a thread memory, only to read a small part of it, degrades badly the memory access performance 10-9 up to a factor of 32. The case AoS with S IZE = 0 packs 6 val- ues and will then have a memory performance of 1/6 ≈ 16% compared to SoA, and the highest values of S IZE will display -10 execution time10 (s) / particle number a performance down to 1/32 ≈ 3%. This is shown in Fig. 8. As 100 102 104 106 108 a result, one can clearly see that the performance per particle particle number saturates for a sufficiently large number of particles, with SoA pattern outperforming the AoS with S IZE = 32 by a factor of Figure 7: Benchmark comparing the execution time of AoS vs SoA implemen- 20 and the AoS with S IZE = 0 by a factor of 2. For small par- tation of the Eulerian update step in single precision. ticle numbers, performance is hindered by a less effective usage 90 of memory, additional to the uncoalesced access pattern, as can be seen in Figure 8. 80 AoS0 It is also noticeable that the performance of SoA is slightly AoS32 70 worse than AoS for small particle numbers (up to 1000). This SoA can be attributed to the fact that when the number of particles is 60 small enough, the L1 cache and the threads registers are large enough to keep the whole particles close in memory, hence al- 50 lowing faster access to other position and velocity components 40 for successive operations, while the SoA pattern has to make 30

a request to global memory for every needed data. Neverthe- L1 cache hit rate less, this effect only brings advantage when the particle number 20 is small. When this number increases, the cache cannot hold the data anymore and so that the global memory is used and 10 another long latency fetch has to be performed. The caching 0 advantage is thus eventually taken over by the poor memory ac- 100 102 104 106 108 cess performance, and the crossing between SoA and AoS (with particle number S IZE = 0) occurs around 2000 particles. This corresponds to a full utilisation of the L1 cache which is 48 kB, the size of one Figure 8: L1 cache hit rate for global memory load requests, in percents. SoA (S IZE = 0) particle being 6 ∗ 4 = 24 bytes. We also performed this simple benchmark on another multi- purpose graphic card boarded on a desktop computer. For this C++ because is enables powerful mechanisms to build abstrac- example, we used the Nvidia Geforce GT755M, composed of tions without loss of performance. We discussed in the intro- 384 cores on 2 multiprocessors, with ∼ 262 kbytes. Timings duction that adding a property (such as a charge) to a particle were 2 − 3 times slower, irrespective of the number of particles requires the modification of all member functions (such as Par- and both for AoS and SoA (not shown here), illustrating the tArr::allocate) that handle the different arrays. C++ allows to benefit of using a graphic card specifically dedicated to high pass this task to the compiler. Using template meta program- performance computing exhibiting more parallelism. ming [6], the needed code can be automatically generated dur- ing the compilation. The result is a class that contains an array for each particle property, the associated access functions and 4. Generic C++ implementation of a structure of arrays member functions that allow efficient handling of all arrays. (SoAx) 4.1. Using SoAx In the introduction we have seen that implementing, main- taining and using a structure of array can be annoying. We Before discussing details of the implementation let us first present now an implementation of a structure of array using show a short listing presenting some functionality of SoAx. Let modern C++ (in fact C++11), called SoAx, that provides a handy us assume that we want our particles to have an identity, a posi- interface, high flexibility and optimal performance. We use tion, a velocity, and a mass of types int, double, double, and 6 120 soax . push back( particle );

100 The necessary class from which the particle objects are cre- ated is also automatically created by the compiler by means of 80 template meta programming. This technique will be discussed in the next section. AoS0 ciency

ffi 60 AoS 32 4.2. Implementation of SoAx SoA SoAx uses inheritance in combination with template meta-

Load e 40 programming. The basic idea is to inherit all arrays (parti- cle properties) into one single structure. The different prop- 20 erty types of the particle are passed to the SoAx class using std::tuple. This is a component of C++11 storing heterogeneous 0 0 2 4 6 8 data types. 10 10 10 10 10 A SoAx attribute consists of an array for storing and particle number member-functions for accessing data. We have chosen to generate these attribute classes by macros to avoid repet- Figure 9: Loading efficiency from the main GPU memory. This is the ratio between requested memory and effectively used memory. itive implementations as they have all the same structure. Macros permit to give custom names to the attributes: From SOAX_ATTRIBUTE(pos, ’P’); the compiler creates a class float, respectively. Let us further assume that we need three- with a member-function pos to access individual particles and dimensional coordinates for the position and velocity. Here is posArr to access directly the complete array. The character what one could write using SoAx: P is only a descriptive string that can be used by the user for other purposes. pos is an instantiation of the class Listing 5: Example code showing typical usage of SoAx template holding a N-dimensional array of type double. // Define particle properties through macro Let us here mention that advanced programming techniques SOAX ATTRIBUTE ( id , ’N ’ ) ; // identity can be used to provide usage safety. The dimensionality is for // SOAX ATTRIBUTE ( pos , ’P ’ ) ; position example automatically taken into account for the member func- SOAX ATTRIBUTE ( v e l , ’V ’ ) ; // velocity tion pos. In the case of pos, pos(42,0) gives SOAX ATTRIBUTE ( mass , ’M’ ) ; // velocity the expected access to the first coordinate of particle 42 while // Specify types and dimension and pos(42) yields a compile-time assertion (through ’substitution // concatenate attributes using std :: tuple failure is not an error’ (SFINAE, [7])). The behavior is the op- typedef std :: tuple , posite in the case of id, where id(42) is the identity pos , of particle 42 and id(42,0) results in a compile-time assertion. vel , Advanced programming techniques also allow to enable the mass< float ,1>> ArrayTypes ; library user to write automatically optimized code. The line soax.posArr(0) = soax.velArr(1)-soax.velArr(2); // create SoA for 42 particles in List. 5 performs an operation on all particles. The library < > Soax ArrayTypes soax(42) ; user does not need to write a custom for-loop for CPUs or a // access properties of particle 23 CUDA kernel for GPUs. For this, SoAx uses a technique called soax . id (23) = 0; // set identity expression templates [8, 7] where a computation such as a sum soax .pos(23 ,0) = 100; // set x− coordinate is encoded in a template. Chained arithmetic operations are analysed at compile time and an optimized code without un-

// operations on all particles (x= vy − vz ) necessary copies is generated by the compiler. This technique soax . posArr (0) =soax . velArr (1)−soax . velArr (2) ; is nowadays used in linear algebra software [9].

// allocate memory of 100 particles 4.2.1. Adding functions soax. resize (100); The user can easily add custom functions to SoAx that he We have payed attention to the fact, that user might want wants to be applied to all arrays. For this, it is not necessary to extract and treat particles as objects (in the spirit of to touch the code of the library. The user only has to define a struct Particle). With SoAx one can write structure containing a doIt member-function (see List. 7 for an example). The first template parameter of this member doIt is Listing 6: Example of using SoAx elements a reference to one of the SoA arrays. Other parameters can be auto particle = soax . getElement (7) ; freely chosen (internally SoAx uses variadic templates). Here particle .id() = 42; is an example of a function that sets the values of all arrays to a particle .pos(0) = 3.14; certain value: 7 Listing 7: Example of a function to be applied to all SoAx arrays 4.2.2. GPU implementation struct SetToValue Several restrictions apply when working with GPU proces- { sors. A first one is the costly data transfer between CPU and template< class T, class Type> GPU: one has to design a solution in which those transfers are static void doIt (T& t , Type value ) { minimized. Data should reside mainly on the GPU and be trans- for ( int i =0;i size () ; i++) ferred to the main memory only when needed by the CPU, for t −>operator []( i) = value ; example for output to a hard drive. One thus cannot make use of } solutions that would results in dereferenciation by the CPU of } ; each elements one at a time, but must rely on device functions that process all data at once on the device. In addition, when Passing this function to a SoAx object soax as a template processing multiple vectors with several operations, processing argument, them all together is faster than successively, an optimisation soax.apply(42); applies sometimes referred to as loop fusion. These constraits lead us SetToValue::doIt to all arrays in soax. to make again use of expression templates for device data. This is achieved via recursive templates. We discuss this pro- Another constrain comes from the fact that C++-stl vectors gramming technique here as a showcase for the doIt function are not designed to work on GPU processors within the CUDA as it explains how templates can be used to make the com- framework, as far as the version 7.0, and another type of data piler generate code without loss of performance (see List. 8). storage is then needed. To allow expression templates to work In fact, the SoAx member-function apply calls the member- with GPUs, we build a custom class, called deviceWrapper, en- function doIt of the class template TupleDo with the particle compassing a pointer to data living on the device. In addition, attribute tuple (Tuple), its size (N) and the user defined tem- as the THRUST library provides the best mimic of stl vectors = plate (DoItClass e.g. SetToValue) as template arguments. structure and algorithms to our knowledge, we also keep trace The member-function doIt calls recursively TupleDo::doIt of the associated device vector to allow efficient operations to for the attribute tuple but passing a decremented size. This re- be performed on the data. cursion continues until the passed size is one so that the com- When an assignment (of the form piler chooses the partially specialized case below. Its doIt soax.posArr(0) = soax.velArr(1)-soax.velArr(2);) member-function calls the doIt function of the user provided is performed, a kernel is called and passed a copy of the DoItClass that terminates the treatment of the first entry of underlying deviceWrapper object, accessing the data with the attribute tuple Tuple. After that the DoItClass::doIt is the expression template objects. The copy constructor of the called for the second entry. This process continues for all at- deviceWrapper class then needs to be overloaded in order to tributes. As the code for all calls is generated at compile time, copy only the raw device pointer and not all the data at each there is no performance overhead compared to a hand-written call. code. Fig. 10 shows a benchmark evaluating the performance of Listing 8: Example explaining compile time code creation by recursive tem- this implementation for the operation (1) as a function of the plates particle number, along with the SOA and AOS (with S IZE = template< class Tuple , std :: size t N, class 16) implementations as references. The time is measured this DoItClass> time with a std::chrono rather then with the kernel profiler, al- struct TupleDo { lowing to assess the possible overhead of the SoAx solution. With this benchmark, we confirm that the performance of SoAx template< class . . . Args> is the same as the SOA also on GPUs. Indeed, the SoAx GPU static void doIt ( Tuple& t , Args . . . args ) implementation comes down in fine to call a kernel on the stored { data adressed through expression templates. TupleDo :: doIt (t , args ...) ; DoItClass :: doIt ( std :: get(t),args...) ; 5. Conclusions } } ; The goal of the work is two-fold. First, it shows that het- erogeneous data (such as particles) should be implemented in template< class Tuple , class DoItClass> an array of structure (AoS) fashion rather than in a structure of struct TupleDo { array (SoA) one if performance is crucial. AoS are generally template< class . . . Args> much faster on modern CPUs as well as on GPUs. The reason static void doIt ( Tuple& t , Args . . . args ) is that AoS better uses cache and vectorization resources that { can speed up typical number crunching algorithms on particles DoItClass :: doIt ( std :: get <0>(t),args...) ; by more than one order of magnitude. However, implementing } and maintaining AoS can be cumbersome especially if the the } ; number of numerical types representing a particle change from one application to another. SoA are in general more handy and 8 -5 10 [7] D. Vandevoorde, N. Josuttis, C++ Templates: The Complete Guide, SoAx Addison-Wesley Professional, 2002. SoA [8] T. Veldhuizen, Expression templates, C++ Report 7 (1995) 26–32. -6 10 AoS [9] A. M. Aragon,´ A c++11 implementation of arbitrary-rank tensors for high- 32 performance computing, Computer Physics Communications 185 (2014) 1681 – 1696. 10-7

10-8

10-9

-10 execution time10 (s) / particle number 100 102 104 106 108 particle number

Figure 10: Benchmark of SoAx library GPU implementation.

flexible. This consideration leads to the second contribution of this work showing that modern C++ programming techniques permits to combine the advantages of both concepts (SoA and AoS) to build a generic library that has the performance of SoAs and the flexibility and handiness of AoS. We demonstrate the benefit of template meta programming for scientific codes. This technique delegates code generation to the compiler and allows for highly readable, maintainable and fast application code. The presented library SoAx runs on CPUs as well as on GPUs.

Acknowledgment

We thank K. Thust for useful discussion and his help using the MIC co-processor. We also thank A. Miniussi for fruitful advices concerning Template Meta Programming. Access to supercomputer Jureca and Juropa3 at the FZ Julich¨ was made available through project HBO22. Part of the computations were performed on the ’mesocentre de calcul SIGAMM’ in Nice.

References

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Working in order to gain a PhD diploma is something I have found really, really interesting. At the end of this journey, I personally see a three groups classification in the set of skills a PhD candidate has to develop during his thesis: scientific, technical, and managing. First one encompasses the core of your work. Gathering present-day knowledge about a specific topic. Designing experiments to be performed. Presenting scientific work and results through talks, posters or articles to a variety of audiences: students, peers, scientific from other communities, private investors... There is a shining excitement, and a subtle beauty in all this process. Second group may vary a lot, and is related to the tools you have actually used for your work: building and using variety of top tech machines. I didn’t have the opportunity to do what we call ”experimental work”, but from what I know, the issue is the same as in a ”numerical work”: devil is in the details. You have to be careful about everything, every setup, every screw, every line of code, every possible bug... you have to make sure that the machine that you use indeed does what you want it to do ! Otherwise your results might be irrelevant and you have to start all over again. The third is the human skills that you acquire. Depending on if you had to manage people or not: how to give motivation but mostly how to keep self-motivation on long- lasting projects. I personally did learn a lot from those tasks and I think that this set of skills is what give PhDs a great autonomy and project-based way of thinking and working. I did also spent a lot of time building what I called technical skills. I apologize for that to my close scientific co-workers, but I know that they know that it is unfortunately an increasing need in nowadays competitive society, more oriented toward short-terms profits rather than fundamental research.

What makes the PhD journey even more interesting is that each is shaped differently depending on its environment. Did the candidate get all the necessary funding to success-

159 160 BIBLIOGRAPHY fully carry his / her research, to benefit from an adequate continuous formation, or was he concerned about tight budget management ? Did he had to work mainly alone ? Did he had to manage interns or engineers ? Did he had to, or choose to focus on technical, engineering details or requirements from his lab or investors ? Without doubt, I was lucky enough to benefit from one very enviable working envi- ronment. A (too) nice place to work in, with the Nice Observatory being situated in the heights of Nice and my living place in the very charming town of Menton, surely I was living in a dreamy sunny place. As a mountain lover that I was during my PhD, this allowed me to run countless of kilometres and discover truly amazing places. Sport is something that had a central importance aside of my work charge, allowing a good equilibrium between intellectual and physical loads. Very nice places to break the day: many thanks to the restaurant staff of the observa- tory: Karima, Nadia, Khaled, Ghislain... A good restaurant in a company makes possible fruitful discussions in an enjoyable place and they for sure did a wonderful job throughout those years. The little kitchen in which I’ve spent countless of coffee breaks was also a nice place to bring the needed serenity between work sprints. Nice colleagues and friends: I was very pleased to work with the others PhD students and post-docs I’ve met during my thesis: Mamadou, Christophe S, Christophe P and Christophe H, Judit, Sophia, Simon... And Giorgio Krstulovic as co-advisor, who also initiated me to the art of rock climbing. An amazing thesis director, displaying a rare combination of human, management and scientific skills. Giving responsibility without ceasing to believe in his people in charge. Always patient and ready to help. For all this, he is now one of my primary source of inspiration for my upcoming career. I’ve met some (hopefully not that many) PhD candidates experiencing disastrous professional relationships with their thesis director, and this is truly sad, considering how this factor is one of the most important and determining factor during a PhD thesis. I wish to thank all the administrative staff from the observatory, Nice university and CNRS, especially Rose Pinto, Elizabeth Taffin-de-Givenchy, Monique Clatot, for their help in all administrative stuffs, making them less ever-lasting. A big thank you to many people with which I had very interesting technical and / or scientific dicussions: Fabrice Ubaldi, Uriel and Helene Frisch, Annick Pouquet, Pablo Mininni, Berangere Dubrulle, Guido Boffetta, Stefano Musacchio, Rahul Pandit, Samriddhi Sankar Ray... while I don’t want to miss anyone, I could not cite them all here. I’ve learned a lot from those people who brought me motivation and inspiration. I was also lucky to work and discuss with people from which I did learn a lot about science politics and higher education in France. Those include people from the team of the association for the young researchers of the Maritime Alps, people from the scientific council of the Nice university, and once again my thesis advisor. Thank you to my friends and collaborators in Belgium: Julien, Charles, Michel, my sis- ter Marie, from which I was able to stay up to date about business and human management. BIBLIOGRAPHY 161

Thank you to my family and my beloved Celine. It is never evident for non-researchers to understand the relevance and implication of the work that a PhD candidate has to carry, neither the irregular working loads, countless late evenings that it requires. They were nevertheless very comprehensible, bringing their own personal colouring to my work, and for some of them, always ready for party. The combination of all these factors and people I’ve met these years have resulted in a fantastical experience during which time flew way too fast, and that shaped my brain and my heart in a way I couldn’t have hoped better.